Year: 2025
Type: article
Abstract: Let G be a graph and (T, B) be a tree-decomposition of G, where B = (Bt) t∈V (T ) .The chromatic number of (T, B) is the maximum chromatic number among the subgraphs induced by Bt.The tree-chromatic number of G is the minimum chromatic number of the tree-decompositions of G. Huynh, Reed, Wood, and Yepremyan [2019-20 MATRIX Annals, Springer, Cham, 2021, 489-498] posed a Hadwiger-type conjecture for tree-chromatic number, and asked for a short proof that every K6-minor-free graph has tree-chromatic number at most 5 even if it is allowed to use the Four Color Theorem.The present article answers this question by observing that, assuming Hadwiger's Conjecture for Kp-minor-free graphs, every Kp+1minor-free graph has tree-chromatic number at most p.More precisely, only path-decompositions of graphs are considered.
Source: Discrete Mathematics Letters
Author: Hikaru Yokoi
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Award: G925228835
OpenAlex ID: W4410130746