Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour [JCTB 2016]. In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger’s Conjecture for tree chromatic number. As evidence that our conjecture may be more tractable than Hadwiger’s Conjecture, we give a short proof that every K5-minor-free graph has tree-chromatic number at most 4, which avoids the Four Colour Theorem. We also present some hardness results and conjectures for computing tree- and path chromatic number.
|Title of host publication||2019-20 MATRIX Annals|
|Editors||David R Wood, Jan de Gier, Cheryl E Praeger, Terence Tao|
|Place of Publication||Cham Switzerland|
|Number of pages||10|
|Publication status||Published - 2021|
|Name||MATRIX Book Series|