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. Consider the two inequalities separately. 1 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. Consider the two inequalities separately. 1 t ( G ) ≤ a ( G ) , t ( G ) ≤ c ( G ) Find individual bounds for t ( G ) , a ( G ) and c ( G ) . 2 � 2 m � n , 2 n − 1 ≤ a ( G ) , 2 m − n + 1 ≤ c ( G ) t ( G ) ≤ n Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 3 / 12

The Merino-Welsh conjecture Thomassen’s approach. Consider the two inequalities separately. 1 t ( G ) ≤ a ( G ) , t ( G ) ≤ c ( G ) Find individual bounds for t ( G ) , a ( G ) and c ( G ) . 2 � 2 m � n , 2 n − 1 ≤ a ( G ) , 2 m − n + 1 ≤ c ( G ) t ( G ) ≤ n Theorem (Thomassen, 2010) Let G be a 2-connected loopless multigraph. n = | V ( G ) | , m = | E ( G ) | . t ( G ) ≤ a ( G ) if m ≤ 1 . 066 n, t ( G ) ≤ c ( G ) if m ≥ 4 n − 4 . Seongmin Ok (DTU) The Merino-Welsh conjecture GT2015 3 / 12

The Merino-Welsh conjecture Thomassen’s approach. Consider the two inequalities separately. 1 t ( G ) ≤ a ( G ) , t ( G ) ≤ c ( G ) Find individual bounds for t ( G ) , a ( G ) and c ( G ) . 2 � 2 m � n , 2 n − 1 ≤ a ( G ) , 2 m − n + 1 ≤ c ( G ) t ( G ) ≤ n Theorem (Thomassen, 2010) Let G be a 2-connected loopless multigraph. n = | V ( G ) | , m = | E ( G ) | . t ( G ) ≤ a ( G ) if m ≤ 1 . 066 n, t ( G ) ≤ c ( G ) if m ≥ 4 n − 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 ) c ( G ) t ( G ) = T ( G ; 1 , 1 ) 2 t ( G ) a ( G ) = T ( G ; 2 , 0 ) 1 a ( G ) c ( G ) = T ( G ; 0 , 2 ) − 1 1 2 3 0 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 ) c ( G ) t ( G ) = T ( G ; 1 , 1 ) 2 t ( G ) a ( G ) = T ( G ; 2 , 0 ) 1 a ( G ) c ( G ) = T ( G ; 0 , 2 ) − 1 1 2 3 0 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 ) c ( G ) t ( G ) = T ( G ; 1 , 1 ) 2 t ( G ) a ( G ) = T ( G ; 2 , 0 ) 1 a ( G ) c ( G ) = T ( G ; 0 , 2 ) − 1 1 2 3 0 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 ) c ( G ) t ( G ) = T ( G ; 1 , 1 ) 2 t ( G ) a ( G ) = T ( G ; 2 , 0 ) 1 a ( G ) c ( G ) = T ( G ; 0 , 2 ) − 1 1 2 3 0 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 ) c ( G ) t ( G ) = T ( G ; 1 , 1 ) 2 t ( G ) a ( G ) = T ( G ; 2 , 0 ) 1 a ( G ) c ( G ) = T ( G ; 0 , 2 ) − 1 1 2 3 0 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 x i 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 x i 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