On Spanning Trees with few Branch Vertices Warren Shull Emory - PowerPoint PPT Presentation

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

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

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 2 k + 3 vertices. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 31 / 169

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 2 k + 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

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 2 k + 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

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 2 k + 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 2 k + 3 vertices... Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 34 / 169

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 2 k + 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 2 k + 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

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 2 k + 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 2 k + 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

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 2 k + 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 2 k + 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

k + 1 K m K m K m K m K m K m K m Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 38 / 169

k + 1 K m K m K m K m K m K m K m 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

k + 1 K m K m K m K m K m K m K m 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

k + 1 K m K m K m K m K m K m K m 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

k + 1 K m K m K m K m K m K m K m 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

k + 1 K m K m K m K m K m K m K m 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

k + 1 K m K m K m K m K m K m K m 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

k + 1 K m K m K m K m K m K m K m Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 45 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 46 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 47 / 169

k + 1 K m K m K m K m K m K m K m | 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

k + 1 K m K m K m K m K m K m K m | 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

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 50 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 51 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k independent set Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 52 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k independent set X Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 53 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k X Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 54 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | ≤ Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 55 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | ≤ k + 3 Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 56 / 169

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

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | ≤ k + 3 + k = 2 k + 3 Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 58 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 59 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 � deg( x ) ≥ x ∈ X Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 60 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 � deg( x ) ≥ x ∈ X Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 61 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 � deg( x ) ≥ ( k + 3)( m − 1) x ∈ X Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 62 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 � deg( x ) ≥ ( k + 3)( m − 1) x ∈ X Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 63 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 � deg( x ) ≥ ( k + 3)( m − 1) + 3 k x ∈ X Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 64 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 � deg( x ) ≥ ( k + 3)( m − 1) + 3 k x ∈ X = mk − k + 3 m − 3 + 3 k Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 65 / 169

k + 1 K m K m K m K m K m K m K m | V ( G ) | = m ( k + 3) + 2 k = mk + 3 m + 2 k | X | = k + 3 + k = 2 k + 3 � deg( x ) ≥ ( k + 3)( m − 1) + 3 k x ∈ X = mk − k + 3 m − 3 + 3 k = mk + 3 m + 2 k − 3 Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 66 / 169

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

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 2 k + 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 2 k + 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

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 2 k + 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 2 k + 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

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 2 k + 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

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 2 k + 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

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 2 k + 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

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 2 k + 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

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 2 k + 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

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 2 k + 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

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 2 k + 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 76 / 169

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 2 k + 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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 77 / 169

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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 78 / 169

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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 79 / 169

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): Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 80 / 169

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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 81 / 169

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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 82 / 169

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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 83 / 169

Among spanning trees with at most 6 leaves, choose a tree T such that: Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 84 / 169

Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 85 / 169

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). Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 86 / 169

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 Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 87 / 169

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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 88 / 169

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? Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 89 / 169

First case: T has only 5 leaves (the fewest possible): Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 90 / 169

First case: T has only 5 leaves (the fewest possible): Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 91 / 169

First case: T has only 5 leaves (the fewest possible): path path path path path path path Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 92 / 169

Second and third cases: T has 6 leaves, but only 3 branch vertices. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 93 / 169

Second and third cases: T has 6 leaves, but only 3 branch vertices. path path path path path path path path Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 94 / 169

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 Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 95 / 169

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. Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 96 / 169

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 97 / 169

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) path path path path path path path path path Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 98 / 169

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) path path path path path path path path path (T5) Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 99 / 169

Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves) path path path path P path path path path path (T5) Warren Shull Emory University Joint work with Ron Gould On Spanning Trees with few Branch Vertices May 18, 2019 100 / 169