Three-coloring triangle-free graphs without long forbidden paths

Yidong Zhou; Jorik Jooken; Baoyuan Shan; Jan Goedgebeur; Shenwei Huang

Paper

Three-coloring triangle-free graphs without long forbidden paths

Abstract

A graph

-vertex-critical if

, but

for every proper induced subgraph

. For a family of graphs

-free if no graph

is an induced subgraph of

. We show that there are exactly three 4-vertex-critical

-free graphs containing an induced

, thereby settling the first of the two cases of a conjecture by Goedgebeur and Schaudt [J.~Graph Theory, 87:188--207, 2018]. Moreover, we show that all

-free graphs are

-colorable and by combining our result with known results from the literature, we completely characterize the maximum chromatic number of

-free graphs if

is a six-vertex induced subgraph of

. Finally, we construct an infinite family of

-vertex-critical

-free graphs. These graphs are also

-free and this is the first value of

for which an infinite family of

-vertex-critical

-free graphs is known.