Matroid Tree Width and the Chromatic Polynomial Rhiannon Hall Collaborators: Carolyn Chun, Criel Merino, Steve Noble The idea of tree-width in graphs: To say how “tree-like” the structure of a graph is. tree decomposition The width of a tree decomposition is the highest number of graph vertices contained in a node of the tree, minus one. The idea was extended to matroids by Hlineny and Whittle The idea was extended to matroids by Hlineny and Whittle tree decomposition Tree decomposition: • The nodes of 푇 are “buckets”. • Each element of 푀 goes in one bucket (partitioning 퐸(푀)). Buckets can be empty or contain many elements. Tree-Width for Matroids • Given a tree decomposition, each node gets a weight called its node-width. • The width of a tree decomposition is the highest value of its node-widths. • The tree-width of a matroid is the minimum width of all its tree decompositions. Tree-Width for Matroids • Given a tree decomposition, each node gets a weight called its node-width. • The width of a tree decomposition is the highest value of its node-widths. • The tree-width of a matroid is the minimum width of all its tree decompositions. • How do we measure node-width? The width of a node 푥 of 푇 is 푊 푥 = 푟 푀 − ∑ 푟 푀 − 푟 퐸 푀 − 퐵푖 = ∑푟 퐸 푀 − 퐵푖 − 푑 − 1 푟 푀 . The width of a node 푥 of 푇 is 푊 푥 = 푟 푀 − ∑ 푟 푀 − 푟 퐸 푀 − 퐵푖 = ∑푟 퐸 푀 − 퐵푖 − 푑 − 1 푟 푀 . What does this value mean geometrically? Brief Digression: A note on 푘-separations in representable matroids. Let 푀 be a matroid representable over 퐺퐹(푞). Then we can embed 푀 in the projective space 푃퐺(푟 푀 − 1,푞). Let (퐴, 퐵) be a 푘-separation of 푀. 푐푙 퐴 ∩ 푐푙 퐵 퐴 퐵 “guts” of (퐴, 퐵) plonk in points to get projective space here Then we can add points of 푃퐺(푟 푀 − 1,푞) to 푀 into the guts of (퐴, 퐵) to obtain matroid 푀′ and 푘-separation (퐴′, 퐵′ ) such that the guts of (퐴′, 퐵′) is the projective space 푃퐺(푘 − 2, 푞). The width of a node 푥 of 푇 is 푊 푥 = 푟 푀 − ∑ 푟 푀 − 푟 퐸 푀 − 퐵푖 = ∑푟 퐸 푀 − 퐵푖 − 푑 − 1 푟 푀 . What does this value mean geometrically? If 푀 is representable, then the width of node 푥 is 푊 푥 = 푟푀′ 퐴 ∪ 퐶1 ∪ 퐶2 ∪ ⋯ ∪ 퐶푑 . If 푀 is representable, then the width of node 푥 is 푊 푥 = 푟푀′ 퐴 ∪ 퐶1 ∪ 퐶2 ∪ ⋯ ∪ 퐶푑 . Note: If 푥 is a leaf node of 푇, and 퐴 is the set of matroid elements in the 푥- bucket, then 푊 푥 = 푟푀 퐴 . Example: Tree decompositions for 푈3,10 and 푈7,10 푈3,10 (matroid elements are 푥1, 푥2, … , 푥10) 푥 푥 1 2 푥3 푥4 푥5 푥6 푥7 푥8 푥9 푥10 1 2 3 3 3 3 3 3 2 1 node widths 푈7,10 푥 푥 1 2 푥3 푥4 푥5 푥6 푥7 푥8 푥9 푥10 1 2 3 4 4 4 4 3 2 1 node widths Example: Tree decompositions for 푈3,10 and 푈7,10 푥10 1 푈3,10 푥9 2 푥 푥 1 2 푥3 푥4 푥5 푥6 푥7 푥8 1 2 3 3 3 3 2 1 node widths 푥10 1 푈7,10 푥9 2 푥 푥 1 2 푥3 푥4 푥5 푥6 푥7 푥8 1 2 3 6 4 3 2 1 node widths Example: Tree decompositions for a daisy 3 퐵 3 퐵1 2 ø 퐵3 3 5 퐵5 3 node widths 퐵4 3 Example: Tree decompositions for a daisy 3 퐵 3 퐵1 2 ø 퐵3 3 5 퐵5 3 퐵4 3 퐵 3 퐵1 2 3 ø 3 3 퐵5 node widths ø 3 ø 3 퐵3 3 퐵4 3 The Chromatic Polynomial for Matroids For a matroid M, |푋| 푟 푀 −푟(푋) 휒 푀, 휆 = (−1) 휆 . 푋⊆퐸 푀 The familiar deletion-contraction rules apply: • If 푥 is not a loop or coloop then 휒 푀, 휆 = 휒 푀\푥, 휆 − 휒 푀/푥, 휆 , • If 푥 is a loop then 휒 푀, 휆 = 0, • If 푥 is a coloop then 휒 푀, 휆 = 휆 − 1 휒 푀\푥, 휆 . Theorem (Chun, H, Merino, Noble): Let 푀 be a matroid representable over 퐺퐹(푞). If 푀 has tree-width at most 푘, then there exists 푐푘 ∈ ℝ such that 휒 푀, 휆 > 0 for all 휆 > 푐푘. Proof Outline: For induction, lexicographically order all 퐺퐹 푞 -representable matroids by (rank, |퐸 푀 |, something). We make sure 푐푘 is large enough that 푃퐺 0, 푞 ,푃퐺 1, 푞 ,… , 푃퐺 푘 − 1,푞 satisfy the theorem. Induction: Let 푀 be 퐺퐹 푞 -representable with tree-width ≤ 푘. Assume all 퐺퐹 푞 -representable matroids of tree-width ≤ 푘 that occur before 푀 in the LEX ordering satisfy the theorem. Let 푇 be an optimal tree decomposition of 푀 and let (퐴, 퐵) be a 푘′-separation of 푀 displayed by 푇 (note 푘′ ≤ 푘): Let {푒1, 푒2, … , 푒푡} be the elements of 푃퐺 푟(푀) − 1, 푞 in the guts of 퐴, 퐵 . 푒푖 Let 푀 denote the matroid 푀 extended by 푒푖. Lemma: 푀푒1,푒2,…,푒푡 has the same tree-width as 푀 (construct the same tree decomposition with 푒1, 푒2,… , 푒푡 in bucket 푣푎or 푣푏). Lemma: For any matroid 푀 and any element 푒 ∈ 퐸(푀), 푇푊(푀/e) ≤ 푇푊(푀). Induction: 푒1,푒2,…,푒푡 We need to know that 휒(푀 , 휆) is positive for all 휆 ≥ 푐푘. Lemma: If the guts is a modular flat then 푒1푒2…푒푡 푒1푒2…푒푡 푒 푒 …푒 휒 (푀|퐴) , 휆 휒((푀|퐵) , 휆) 휒 푀 1 2 푡 , 휆 = 휒({푒1, 푒2, … , 푒푡}, 휆) 푒1푒2…푒푡 푒1푒2…푒푡 푒 푒 …푒 휒 (푀|퐴) , 휆 휒((푀|퐵) , 휆) 휒 푀 1 2 푡 , 휆 = 휒({푒1, 푒2, … , 푒푡}, 휆) Rank lower than 푀 so the induction works … … except in the case where 푀 has no 푘′-separations for any 푘′ < 푟 푀 . In which case, extend M to 푃퐺 푟(푀) − 1, 푞 as on the previous slide. The End – Questions? Pictures: Illegal origami & Halloween origami .