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Sep 19, 2022 225 likes •1.94k views

On Spanning Trees with few Branch Vertices Warren Shull Emory University Joint work with Ron Gould May 18, 2019 Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 1 / 169 Spanning

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On Spanning Trees with few Branch Vertices

Warren Shull Emory University Joint work with Ron Gould May 18, 2019

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 1 / 169

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Spanning trees

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 2 / 169

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Spanning trees Every connected graph has one.

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Spanning trees Every connected graph has one.

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Spanning trees Every connected graph has one.

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Spanning trees

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Spanning trees Degree 1: leaf

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 7 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 9 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1)

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 10 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1)

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 11 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1) No branch vertices

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 12 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1) No branch vertices

Some spanning trees are “close” to being a Hamiltonian path, in a few different ways:

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 13 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1) No branch vertices

Some spanning trees are “close” to being a Hamiltonian path, in a few different ways:

Low maximum degree

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 14 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1) No branch vertices

Some spanning trees are “close” to being a Hamiltonian path, in a few different ways:

Low maximum degree Few leaves

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 15 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1) No branch vertices

Some spanning trees are “close” to being a Hamiltonian path, in a few different ways:

Low maximum degree Few leaves Few branch vertices

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 16 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1) No branch vertices

Some spanning trees are “close” to being a Hamiltonian path, in a few different ways:

Low maximum degree Few leaves Few branch vertices

Throughout this talk, spanning trees are “better” the fewer branch vertices they have.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 17 / 169

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Spanning trees Degree 1: leaf Degree ≥ 3: branch vertex Hamiltonian paths are a special kind of spanning tree

Max degree 2 (except K2 and K1) 2 leaves (except K1) No branch vertices

Some spanning trees are “close” to being a Hamiltonian path, in a few different ways:

Low maximum degree Few leaves Few branch vertices

Throughout this talk, spanning trees are “better” the fewer branch vertices they have. What conditions might lead to a good spanning tree?

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 18 / 169

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Independent sets v. spanning trees

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Independent sets v. spanning trees a good spanning tree is reached by adding edges

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Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 21 / 169

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Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges Given the right parameters, one or the other must exist.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 22 / 169

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Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more...

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 23 / 169

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Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Among maximum independent sets, which one has the smallest total degree?

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 24 / 169

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Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Among maximum independent sets, which one has the smallest total degree?

If we remove enough edges, we can make this number smaller!

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 25 / 169

SLIDE 26

Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Among maximum independent sets, which one has the smallest total degree?

If we remove enough edges, we can make this number smaller!

Given the right parameters, there is either a good spanning tree or a large independent set with small total degree.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 26 / 169

SLIDE 27

Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Among maximum independent sets, which one has the smallest total degree?

If we remove enough edges, we can make this number smaller!

Given the right parameters, there is either a good spanning tree or a large independent set with small total degree. And of course, it also helps if the graph is claw-free.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 27 / 169

SLIDE 28

Independent sets v. spanning trees a good spanning tree is reached by adding edges a large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Among maximum independent sets, which one has the smallest total degree?

If we remove enough edges, we can make this number smaller!

Given the right parameters, there is either a good spanning tree or a large independent set with small total degree. And of course, it also helps if the graph is claw-free. What are the best possible parameters?

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 28 / 169

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Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 29 / 169

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Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices,

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 30 / 169

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Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 31 / 169

SLIDE 32

Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 32 / 169

SLIDE 33

Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices,

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 33 / 169

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Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices...

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 34 / 169

SLIDE 35

Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 35 / 169

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Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 36 / 169

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Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 37 / 169

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Km Km Km Km Km Km Km k + 1

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 38 / 169

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Km Km Km Km Km Km Km k + 1 Connected and claw-free

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 39 / 169

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Km Km Km Km Km Km Km k + 1 Connected and claw-free

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 40 / 169

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Km Km Km Km Km Km Km k + 1 Connected and claw-free Any spanning tree must have a branch vertex in this triangle...

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 41 / 169

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Km Km Km Km Km Km Km k + 1 Connected and claw-free Any spanning tree must have a branch vertex in this triangle...

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 42 / 169

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Km Km Km Km Km Km Km k + 1 Connected and claw-free Any spanning tree must have a branch vertex in this triangle... ...and each of these others...

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 43 / 169

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Km Km Km Km Km Km Km k + 1 Connected and claw-free Any spanning tree must have a branch vertex in this triangle... ...and each of these others... ...for a minimum of k + 1 branch vertices.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 44 / 169

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Km Km Km Km Km Km Km k + 1

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 45 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| =

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 46 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| =

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 47 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3)

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 48 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3)

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 49 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 50 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 51 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k independent set

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 52 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k independent set X

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 53 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k X

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 54 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| ≤

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 55 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| ≤ k + 3

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 56 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| ≤ k + 3 + k

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 57 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| ≤ k + 3 + k = 2k + 3

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 58 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 59 / 169

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Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 60 / 169

SLIDE 61

Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 61 / 169

SLIDE 62

Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥ (k + 3)(m − 1)

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 62 / 169

SLIDE 63

Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥ (k + 3)(m − 1)

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 63 / 169

SLIDE 64

Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥ (k + 3)(m − 1) + 3k

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 64 / 169

SLIDE 65

Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥ (k + 3)(m − 1) + 3k = mk − k + 3m − 3 + 3k

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 65 / 169

SLIDE 66

Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥ (k + 3)(m − 1) + 3k = mk − k + 3m − 3 + 3k = mk + 3m + 2k − 3

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 66 / 169

SLIDE 67

Km Km Km Km Km Km Km k + 1 |V (G)| = m(k + 3) + 2k = mk + 3m + 2k |X| = k + 3 + k = 2k + 3

x∈X

deg(x) ≥ (k + 3)(m − 1) + 3k = mk − k + 3m − 3 + 3k = mk + 3m + 2k − 3 = |V (G)| − 3

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 67 / 169

SLIDE 68

Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 68 / 169

SLIDE 69

Theorem (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 69 / 169

SLIDE 70

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 70 / 169

SLIDE 71

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 71 / 169

SLIDE 72

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Corollary

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 leaves, or an independent set of 3 vertices with at most |V (G)| − 3 outgoing edges.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 72 / 169

SLIDE 73

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Corollary

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 leaves (0 branch vertices), or an independent set of 3 vertices with at most |V (G)| − 3 outgoing edges.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 73 / 169

SLIDE 74

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 74 / 169

SLIDE 75

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Matsuda, Ozeki, Yamashita 2012)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 1 branch vertex, or an independent set of 5 vertices with at most |V (G)| − 3 outgoing edges.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 75 / 169

SLIDE 76

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

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SLIDE 77

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing

edges. This is best possible.

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Gould, S. 2017)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges.

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SLIDE 78

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Gould, S. 2017)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges.

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SLIDE 79

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Gould, S. 2017)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges.

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SLIDE 80

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Gould, S. 2017)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline):

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SLIDE 81

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Gould, S. 2017)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline): Let G be a connected claw-free graph.

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SLIDE 82

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Gould, S. 2017)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline): Let G be a connected claw-free graph. By contradiciton, assume G has neither a spanning tree with at most 2 branch vertices, nor an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges.

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SLIDE 83

Theorem (Kano, et. al. 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G)| − k − 3.

Theorem (Gould, S. 2017)

Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline): Let G be a connected claw-free graph. By contradiciton, assume G has neither a spanning tree with at most 2 branch vertices, nor an independent set of 7 vertices with at most |V (G)| − 3 outgoing edges. By the theorem of Kano et. al. above (with k = 4), G has a spanning tree with at most 6 leaves.

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SLIDE 84

Among spanning trees with at most 6 leaves, choose a tree T such that:

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SLIDE 85

Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible.

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SLIDE 86

Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1).

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SLIDE 87

Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA

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SLIDE 88

Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA (T4) The parts of T in-between branch vertices are as small as possible.

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SLIDE 89

Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA (T4) The parts of T in-between branch vertices are as small as possible. What different structures could T possibly have?

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SLIDE 90

First case: T has only 5 leaves (the fewest possible):

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SLIDE 91

First case: T has only 5 leaves (the fewest possible):

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SLIDE 92

First case: T has only 5 leaves (the fewest possible): path path path path path path path

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SLIDE 93

Second and third cases: T has 6 leaves, but only 3 branch vertices.

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SLIDE 94

Second and third cases: T has 6 leaves, but only 3 branch vertices. path path path path path path path path

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SLIDE 95

Second and third cases: T has 6 leaves, but only 3 branch vertices. path path path path path path path path path path path path path path path path

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SLIDE 96

Second and third cases: T has 6 leaves, but only 3 branch vertices. path path path path path path path path path path path path path path path path (T3) If choosing between trees of these two types, we always choose one of the first type.

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SLIDE 97

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)

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SLIDE 98

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) path path path path path path path path path

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SLIDE 99

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) path path path path path path path path path (T5)

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SLIDE 100

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) P path path path path path path path path path (T5)

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SLIDE 101

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) P path path path path path path path path path (T5) P is as short as possible.

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SLIDE 102

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) P path path path path path path path path path (T5) P is as short as possible. path path path path path path path path path

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SLIDE 103

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

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SLIDE 104

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

Theorem (Gould, S. 2018+)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

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SLIDE 105

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

Theorem (Gould, S. 2018+)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline):

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SLIDE 106

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

Theorem (Gould, S. 2018+)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline): Let G be a connected claw-free graph.

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SLIDE 107

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

Theorem (Gould, S. 2018+)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline): Let G be a connected claw-free graph. By contradiction, assume G has neither the spanning tree nor the independent set with the conditions mentioned above.

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SLIDE 108

Conjecture (Matsuda, Ozeki, Yamashita 2012)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges.

Theorem (Gould, S. 2018+)

Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 3 outgoing edges. Proof (outline): Let G be a connected claw-free graph. By contradiction, assume G has neither the spanning tree nor the independent set with the conditions mentioned above. Choose a spanning tree with the fewest possible branch vertices.

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SLIDE 109

How do we avoid having too many outgoing edges?

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SLIDE 110

How do we avoid having too many outgoing edges? We chose a spanning tree with the fewest branch vertices.

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SLIDE 111

How do we avoid having too many outgoing edges? We chose a spanning tree with the fewest branch vertices. If we find one with fewer, that’s a contradiction.

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SLIDE 112

How do we avoid having too many outgoing edges? We chose a spanning tree with the fewest branch vertices. If we find one with fewer, that’s a contradiction. Some combinations of edges can be ruled out this way, and some neighbor sets can be kept disjoint, which will bound their sum total.

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SLIDE 113

How do we avoid having too many outgoing edges? We chose a spanning tree with the fewest branch vertices. If we find one with fewer, that’s a contradiction. Some combinations of edges can be ruled out this way, and some neighbor sets can be kept disjoint, which will bound their sum total. If a direct overlap of sets will not give a contradiction, sometimes a “shifted” overlap will. But how do we keep the shift consistent?

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SLIDE 114

How do we avoid having too many outgoing edges? We chose a spanning tree with the fewest branch vertices. If we find one with fewer, that’s a contradiction. Some combinations of edges can be ruled out this way, and some neighbor sets can be kept disjoint, which will bound their sum total. If a direct overlap of sets will not give a contradiction, sometimes a “shifted” overlap will. But how do we keep the shift consistent?

Our methods in the previous proof definitely do not generalize!

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SLIDE 115

How do we avoid having too many outgoing edges? We chose a spanning tree with the fewest branch vertices. If we find one with fewer, that’s a contradiction. Some combinations of edges can be ruled out this way, and some neighbor sets can be kept disjoint, which will bound their sum total. If a direct overlap of sets will not give a contradiction, sometimes a “shifted” overlap will. But how do we keep the shift consistent?

Our methods in the previous proof definitely do not generalize!

Our approach to this dilemma is to look at the edges of the spanning tree, of which there are always |V (G)| − 1.

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SLIDE 116

Important idea #1: “Oblique Neighbors”

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SLIDE 117

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors.

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SLIDE 118

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 119

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 120

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 121

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 122

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 123

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 124

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 125

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 126

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 127

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. w x y z

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SLIDE 128

Important idea #1: “Oblique Neighbors” A vertex and a spanning tree edge can be oblique neighbors. Every spanning tree edge has both its incident vertices as oblique neighbors. w x y z

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SLIDE 129

w x y z

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SLIDE 130

Important idea #2: “Pseudoadjacency” w x y z

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SLIDE 131

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 132

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 133

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 134

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 135

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 136

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 137

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 138

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 139

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 140

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 141

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 142

Important idea #2: “Pseudoadjacency” Two vertices are pseudoadjacent if any edge has them both as oblique neighbors. w x y z

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SLIDE 143

How does pseudoadjacency help us control total degree?

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SLIDE 144

How does pseudoadjacency help us control total degree? The degree of a vertex (in the graph, not the tree) equals the number

f tree edges that have it as an oblique neighbor.

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SLIDE 145

How does pseudoadjacency help us control total degree? The degree of a vertex (in the graph, not the tree) equals the number

f tree edges that have it as an oblique neighbor.

Therefore a pseudoindependent set has degrees summing to at most n − 1.

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SLIDE 146

How does pseudoadjacency help us control total degree? The degree of a vertex (in the graph, not the tree) equals the number

f tree edges that have it as an oblique neighbor.

Therefore a pseudoindependent set has degrees summing to at most n − 1. We need only find two tree edges with no oblique neighbor in the set, and this sum drops to n − 3.

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SLIDE 147

FUTURE WORK

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SLIDE 148

FUTURE WORK Km Km Km Km Km Km Km k + 1

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SLIDE 149

FUTURE WORK Km Km Km Km Km Km Km k + 1 In this example, you may notice lots of vertices with degree only 3.

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SLIDE 150

FUTURE WORK Km Km Km Km Km Km Km k + 1 In this example, you may notice lots of vertices with degree only 3. What happens if we impose a higher minimum degree on G?

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SLIDE 151

FUTURE WORK

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SLIDE 152

FUTURE WORK What if we impose a minimum degree of t ≥ 4?

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SLIDE 153

FUTURE WORK What if we impose a minimum degree of t ≥ 4? Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 k + 1

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SLIDE 154

FUTURE WORK What if we impose a minimum degree of t ≥ 4? Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 k + 1 Largest independent set is still 2k + 3

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SLIDE 155

FUTURE WORK What if we impose a minimum degree of t ≥ 4? Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 k + 1 Largest independent set is still 2k + 3 Degrees add up to at least n − 2k − 3

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SLIDE 156

FUTURE WORK What if we impose a minimum degree of t ≥ 4? Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 Kt+1 k + 1

Conjecture

Let k be a positive integer and let G be a connected claw-free graph with minimum degree at least 4. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G)| − 2k − 3 outgoing edges.

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SLIDE 157

FUTURE WORK

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SLIDE 158

FUTURE WORK What if we omit the claw-free condition?

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SLIDE 159

FUTURE WORK What if we omit the claw-free condition? Km Km Km Km Km Km Km k + 1

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SLIDE 160

FUTURE WORK What if we omit the claw-free condition? Km Km Km Km Km Km Km k + 1

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1.

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SLIDE 161

FUTURE WORK What if we omit the claw-free condition? Km Km Km Km Km Km Km k + 1

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1. This is open for all positive k. . .

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 161 / 169

SLIDE 162

FUTURE WORK What if we omit the claw-free condition? Km Km Km Km Km Km Km k + 1

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1. This is open for all positive k but false for k = 0.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 162 / 169

SLIDE 163

FUTURE WORK What if we omit the claw-free condition? Km Km Km Km Km Km Km k + 1

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1. This is open for all positive k but false for k = 0 (consider K⌈ n+2

2 ⌉,⌊ n−2 2 ⌋). Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 163 / 169

SLIDE 164

FUTURE WORK What if we omit the claw-free condition?

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1. This is open for all positive k but false for k = 0 (consider K⌈ n+2

2 ⌉,⌊ n−2 2 ⌋). Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 164 / 169

SLIDE 165

FUTURE WORK

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 165 / 169

SLIDE 166

FUTURE WORK

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 166 / 169

SLIDE 167

FUTURE WORK

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1.

Conjecture

Let k be a non-negative integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1, or an independent set larger than k + 3 vertices.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 167 / 169

SLIDE 168

FUTURE WORK

Conjecture

Let k be a positive integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1.

Conjecture

Let k be a non-negative integer and let G be a connected graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of k + 3 vertices with total degree at most |V (G)| − k − 1, or an independent set larger than k + 3 vertices.

Conjecture (?)

Let k be a non-negative integer and let G be a connected graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices with total degree at most |V (G)| − 1, or an independent set larger than k + 3 vertices.

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 168 / 169

SLIDE 169

Thank you for your attention! wshull@emory.edu mathcs.emory.edu/∼wshull/

Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 169 / 169