Three-coloring triangle-free graphs without long forbidden paths

TL;DR

This paper characterizes certain critical graphs and coloring properties in triangle-free graphs with forbidden paths, settling conjectures and constructing infinite families of critical graphs.

Contribution

It proves the uniqueness of three 4-vertex-critical graphs with specific forbidden subgraphs and characterizes the chromatic number for classes of graphs with forbidden induced subgraphs.

Findings

Exactly three 4-vertex-critical {P_7,C_3}- free graphs with an induced {C_7}.

All {P_5+P_1,C_3}- free graphs are 3-colorable.

Constructed an infinite family of 4-vertex-critical {4K_2,C_3}- free graphs.

Abstract

A graph $G$ is $k$-vertex-critical if $\chi(G)=k$, but $\chi(G')<k$ for every proper induced subgraph $G'$ of $G$. For a family of graphs $\mathcal{F}$, $G$ is $\mathcal{F}$-free if no graph $F \in \mathcal{F}$ is an induced subgraph of $G$. We show that there are exactly three 4-vertex-critical $\{P_7,C_3\}$-free graphs containing an induced $C_7$, 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 $\{P_5+P_1,C_3\}$-free graphs are $3$-colorable and by combining our result with known results from the literature, we completely characterize the maximum chromatic number of $\{F,C_3\}$-free graphs if $F$ is a six-vertex induced subgraph of $P_7$. Finally, we construct an infinite family of $4$-vertex-critical $\{4K_2,C_3\}$-free graphs. These graphs are also $\{P_{11},C_3\}$-free and this is the first value of $t$ for which an infinite family of $4$-vertex-critical $\{P_{t},C_3\}$-free graphs is known.