The Merino-Welsh conjecture Seongmin Ok Department of Applied - - PowerPoint PPT Presentation

The Merino-Welsh conjecture

Seongmin Ok

Department of Applied Mathematics and Computer Science Technical University of Denmark (DTU)

Graph Theory 2015, Nyborg, Denmark

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 1 / 12

Three counting problems

Spanning trees We consider multigraphs, may have parallel edges. but no loops

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 2 / 12

Three counting problems

Spanning trees Acyclic orientations An orientation, without any directed cycle.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 2 / 12

Three counting problems

Spanning trees Acyclic orientations Totally cyclic orientations An orientation, in which every edge is in a directed cycle.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 2 / 12

Three counting problems

Spanning trees Acyclic orientations Totally cyclic orientations t(G) the number of spanning trees a(G) the number of acyclic orientations c(G) the number of totally cyclic orientations

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 2 / 12

Three counting problems

Spanning trees Acyclic orientations Totally cyclic orientations t(G) the number of spanning trees a(G) the number of acyclic orientations c(G) the number of totally cyclic orientations

Conjecture (The Merino-Welsh conjecture, 1999)

If G is a 2-connected loopless multigraph, then t(G) ≤ max{a(G), c(G)}.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 2 / 12

The Merino-Welsh conjecture

Thomassen’s approach.

1

Consider the two inequalities separately. t(G) ≤ a(G), t(G) ≤ c(G)

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 3 / 12

The Merino-Welsh conjecture

Thomassen’s approach.

1

Consider the two inequalities separately. t(G) ≤ a(G), t(G) ≤ c(G)

2

Find individual bounds for t(G), a(G) and c(G). t(G) ≤ 2m

n

n, 2n−1 ≤ a(G), 2m−n+1 ≤ c(G)

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 3 / 12

The Merino-Welsh conjecture

Thomassen’s approach.

1

Consider the two inequalities separately. t(G) ≤ a(G), t(G) ≤ c(G)

2

Find individual bounds for t(G), a(G) and c(G). t(G) ≤ 2m

n

n, 2n−1 ≤ a(G), 2m−n+1 ≤ c(G)

Theorem (Thomassen, 2010)

Let G be a 2-connected loopless multigraph. n = |V(G)|, m = |E(G)|. t(G) ≤ a(G) if m ≤ 1.066n, t(G) ≤ c(G) if m ≥ 4n − 4.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 3 / 12

The Merino-Welsh conjecture

Thomassen’s approach.

1

Consider the two inequalities separately. t(G) ≤ a(G), t(G) ≤ c(G)

2

Find individual bounds for t(G), a(G) and c(G). t(G) ≤ 2m

n

n, 2n−1 ≤ a(G), 2m−n+1 ≤ c(G)

Theorem (Thomassen, 2010)

Let G be a 2-connected loopless multigraph. n = |V(G)|, m = |E(G)|. t(G) ≤ a(G) if m ≤ 1.066n, t(G) ≤ c(G) if m ≥ 4n − 4.

Theorem (Ok, 2012)

t(G) ≤ a(G) if m ≤ 1.29(n − 1), t(G) ≤ c(G) if m ≥ 3.58(n − 1) and G is 3-edge-connected.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 3 / 12

The Merino-Welsh conjecture

Proof idea : flip an edge.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 4 / 12

The Merino-Welsh conjecture

Proof idea : flip an edge. An edge is flippable in an acyclic orientation if flipping preserves the acyclicity.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 4 / 12

The Merino-Welsh conjecture

There are many flippable edges. Lemma 1. (Fukuda et al.) An acyclic orientation of a n-vertex simple connected graph has at least n − 1 flippable edges.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 5 / 12

The Merino-Welsh conjecture

There are many flippable edges. Lemma 1. (Fukuda et al.) An acyclic orientation of a n-vertex simple connected graph has at least n − 1 flippable edges. Lemma 2. A totally cyclic orientation of a 3-edge-connected graph has at least m − n + 1 flippable edges.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 5 / 12

The Merino-Welsh conjecture

There are many flippable edges. Lemma 1. (Fukuda et al.) An acyclic orientation of a n-vertex simple connected graph has at least n − 1 flippable edges. Lemma 2. A totally cyclic orientation of a 3-edge-connected graph has at least m − n + 1 flippable edges. Every totally cyclic orientation has many flippable edges. An edge is flippable in many totally cyclic orientations.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 5 / 12

The Merino-Welsh conjecture

There are many flippable edges. Lemma 1. (Fukuda et al.) An acyclic orientation of a n-vertex simple connected graph has at least n − 1 flippable edges. Lemma 2. A totally cyclic orientation of a 3-edge-connected graph has at least m − n + 1 flippable edges. Every totally cyclic orientation has many flippable edges. An edge is flippable in many totally cyclic orientations. e is flippable iff the orientation is totally cyclic if we remove e.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 5 / 12

The Merino-Welsh conjecture

There are many flippable edges. Lemma 1. (Fukuda et al.) An acyclic orientation of a n-vertex simple connected graph has at least n − 1 flippable edges. Lemma 2. A totally cyclic orientation of a 3-edge-connected graph has at least m − n + 1 flippable edges. Every totally cyclic orientation has many flippable edges. An edge is flippable in many totally cyclic orientations. e is flippable iff the orientation is totally cyclic if we remove e. c(G − e) is close to c(G). c(G) = c(G − e) + c(G/e). c(G) >> c(G/e).

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 5 / 12

The convex version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The three values are evaluations of the Tutte polynomial, T(G; x, y) t(G) = T(G; 1, 1) a(G) = T(G; 2, 0) c(G) = T(G; 0, 2)

−1 1 2 3 1 2

c(G) t(G) a(G)

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 6 / 12

The convex version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The three values are evaluations of the Tutte polynomial, T(G; x, y) t(G) = T(G; 1, 1) a(G) = T(G; 2, 0) c(G) = T(G; 0, 2)

−1 1 2 3 1 2

c(G) t(G) a(G)

A stronger conjecture : t(G) ≤ (a(G) + c(G))/2.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 6 / 12

The convex version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The three values are evaluations of the Tutte polynomial, T(G; x, y) t(G) = T(G; 1, 1) a(G) = T(G; 2, 0) c(G) = T(G; 0, 2)

−1 1 2 3 1 2

c(G) t(G) a(G)

A stronger conjecture : t(G) ≤ (a(G) + c(G))/2. Even stronger conjecture : T(G; x, y) convex on the line segment.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 6 / 12

The convex version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The three values are evaluations of the Tutte polynomial, T(G; x, y) t(G) = T(G; 1, 1) a(G) = T(G; 2, 0) c(G) = T(G; 0, 2)

−1 1 2 3 1 2

c(G) t(G) a(G)

A stronger conjecture : t(G) ≤ (a(G) + c(G))/2. Even stronger conjecture : T(G; x, 2 − x) convex for x ∈ [0, 2].

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 6 / 12

The convex version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The three values are evaluations of the Tutte polynomial, T(G; x, y) t(G) = T(G; 1, 1) a(G) = T(G; 2, 0) c(G) = T(G; 0, 2)

−1 1 2 3 1 2

c(G) t(G) a(G)

A stronger conjecture : t(G) ≤ (a(G) + c(G))/2. Even stronger conjecture : T(G; x, 2 − x) convex for x ∈ [0, 2].

Theorem (Merino et al., 2011)

If M is a coloopless paving matroid, then T(M; x, 2 − x) convex for x ∈ [0, 2]. It is conjectured that almost all matroids are paving matroids.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 6 / 12

The convex version

Theorem (Ok, 2015)

If G be a minimally 2-edge-connected graph, then T(G; x, 2 − x) is convex for x ∈ [0, 2]. Actually, T(G; x, y) is convex on any line with slope −1 in the positive quadrant, since T(G; x, y) is a positive sum of xi and (x + y)j.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 7 / 12

The convex version

Theorem (Ok, 2015)

If G be a minimally 2-edge-connected graph, then T(G; x, 2 − x) is convex for x ∈ [0, 2]. Actually, T(G; x, y) is convex on any line with slope −1 in the positive quadrant, since T(G; x, y) is a positive sum of xi and (x + y)j. Proof for coloopless paving matroids. If possible, T(G) = T(G − e) + T(G/e). Otherwise, either e is a loop or G is minimally 2-edge-connected. We know the full structure when G has a loop.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 7 / 12

Multiplicative version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The convex version is to consider the arithmetic mean. The multiplicative version : t(G)2 ≤ a(G)c(G).

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 8 / 12

Multiplicative version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The convex version is to consider the arithmetic mean. The multiplicative version : t(G)2 ≤ a(G)c(G).

Theorem (Noble and Royle, 2013)

If G is a 2-connected series-parallel graph, then t(G)2 ≤ a(G)c(G).

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 8 / 12

Multiplicative version

The Merino-Welsh conjecture : t(G) ≤ max{a(G), c(G)}. The convex version is to consider the arithmetic mean. The multiplicative version : t(G)2 ≤ a(G)c(G).

Theorem (Noble and Royle, 2013)

If G is a 2-connected series-parallel graph, then t(G)2 ≤ a(G)c(G). A graph is series-parallel if it can be constructed from a single edge by serial and parallel extensions. serial extension parallel extension

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 8 / 12

Multiplicative version

We can keep the initial two vertices during series-parallel extensions. t1 t2 t1 t2 t1 t2 They are called two-terminal series-parallel (ttsp) graphs.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 9 / 12

Multiplicative version

We can keep the initial two vertices during series-parallel extensions. t1 t2 t1 t2 t1 t2 They are called two-terminal series-parallel (ttsp) graphs. All series-parallel graphs are obtained from ttsp graphs by serial and parallel operations. t1 t2 t1 t2 t1 t2 t1 t2 G1 G2 G1 G2 G1 G2

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 9 / 12

Multiplicative version

Noble and Royle’s idea. In some cases, two ttsp graphs are ‘comparable’ in the following sense. H H G1 G2

t(G1 + H)2 ≤ a(G1 + H)c(G1 + H) t(G2 + H)2 ≤ a(G2 + H)c(G2 + H)

regardless of H.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 10 / 12

Multiplicative version

Noble and Royle’s idea. In some cases, two ttsp graphs are ‘comparable’ in the following sense. H H G1 G2

t(G1 + H)2 ≤ a(G1 + H)c(G1 + H) t(G2 + H)2 ≤ a(G2 + H)c(G2 + H)

regardless of H. They found 19 ttsp graphs that replace all others, and confirmed the conjecture.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 10 / 12

Multiplicative version

Consider the graphs of bounded treewidth.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 11 / 12

Multiplicative version

Consider the graphs of bounded treewidth. they have natrual construction analagous to ttsp graphs.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 11 / 12

Multiplicative version

Consider the graphs of bounded treewidth. they have natrual construction analagous to ttsp graphs. I found the replacement condition for arbitrary cut size.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 11 / 12

Multiplicative version

Consider the graphs of bounded treewidth. they have natrual construction analagous to ttsp graphs. I found the replacement condition for arbitrary cut size. A finite list of possible counterexamples for pathwidth 3.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 11 / 12

Multiplicative version

Consider the graphs of bounded treewidth. they have natrual construction analagous to ttsp graphs. I found the replacement condition for arbitrary cut size. A finite list of possible counterexamples for pathwidth 3. more than 5000 graphs.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 11 / 12

Multiplicative version

Consider the graphs of bounded treewidth. they have natrual construction analagous to ttsp graphs. I found the replacement condition for arbitrary cut size. A finite list of possible counterexamples for pathwidth 3. more than 5000 graphs. And, the conjecture holds for graphs of pathwidth 3.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 11 / 12

What next?

Recall that we considered 2T(G; 1, 1) ≤ T(G; 2, 0) + T(G; 0, 2).

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 12 / 12

What next?

Recall that we considered 2T(G; 1, 1) ≤ T(G; 2, 0) + T(G; 0, 2). Put P(t) = T(G; 1 + t, 1 − t), and it is [P(t) + P(−t)]t=0 ≤ [P(t) + P(−t)]t=1 .

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 12 / 12

What next?

Recall that we considered 2T(G; 1, 1) ≤ T(G; 2, 0) + T(G; 0, 2). Put P(t) = T(G; 1 + t, 1 − t), and it is [P(t) + P(−t)]t=0 ≤ [P(t) + P(−t)]t=1 . Is P(t) + P(−t) increasing in t ∈ [0, 1]?

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 12 / 12

What next?

Recall that we considered 2T(G; 1, 1) ≤ T(G; 2, 0) + T(G; 0, 2). Put P(t) = T(G; 1 + t, 1 − t), and it is [P(t) + P(−t)]t=0 ≤ [P(t) + P(−t)]t=1 . Is P(t) + P(−t) increasing in t ∈ [0, 1]? −2t12 −76t10 +184t8 +2836t6 +7762t4 +12960t2 +7776 −2t12 +4t10 −80t10 +160t8 +24t8 +2836t6 +7762t4 +12960t2 +7776 P(t)P(−t) has the same property for small graphs.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 12 / 12

What next?

Recall that we considered 2T(G; 1, 1) ≤ T(G; 2, 0) + T(G; 0, 2). Put P(t) = T(G; 1 + t, 1 − t), and it is [P(t) + P(−t)]t=0 ≤ [P(t) + P(−t)]t=1 . Is P(t) + P(−t) increasing in t ∈ [0, 1]? −2t12 −76t10 +184t8 +2836t6 +7762t4 +12960t2 +7776 −2t12 +4t10 −80t10 +160t8 +24t8 +2836t6 +7762t4 +12960t2 +7776 P(t)P(−t) has the same property for small graphs. I have a (near) proof that if P(t) + P(−t) is, then so is P(t)P(−t). P(t) + P(−t) increasing implies the original conjecture, and P(t)P(−t) increasing implies the multiplicative version. I can only hope for.

Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 12 / 12