On Triangle-Free Graphs That Do Not Contain a Subdivision of the Complete Graph on Four Vertices as an Induced Subgraph

Abstract

International audienc

Triangle‐free graphs that do not contain an induced subdivision of K4 are 3‐colorable

“…Using Theorem , we can prove a variant of a conjecture from that we now explain. For a graph and , we say that is a noncenter pair for if neither nor is the center of a proper wheel in , and or .…”

“…We call a vertex proper for if either ; or all neighbors of in are in one sector of ; and if has more than two neighbors in , then is adjacent to . The wheel is proper if every vertex of is proper for . (Please note that this definition is different from the one in .) We prove…”

“…An is a graph that is isomorphic to a subdivision of . In , two of us studied the structure of ‐free graphs and proposed the following conjecture (and proved several special cases of it):…”

Chromatic Number of ISK4-Free Graphs

“…Every {ISK4, wheel}free graph is 3-colorable. Theorem 1.2 (Trotignon, Vušković 2016 [9]). Every ISK4-free graph of girth at least 5 contains a vertex of degree at most 2 and is 3-colorable.…”

“…Every {ISK4, K 3,3 , prism, K 2,2,2 }free graph contains a vertex of degree at most three. Conjecture 1.4 (Trotignon, Vušković '2016 [9]). Every {ISK4, K 3,3 , triangle}free graph contains a vertex of degree at most two.…”