Coloring graphs with two odd cycle lengths

TL;DR

This paper determines the chromatic number of graphs with exactly two specified odd cycle lengths, providing complete solutions for certain cases and improving classical bounds on graph coloring.

Contribution

The paper completely characterizes the chromatic number for graphs with two specific odd cycle lengths, extending previous results and refining classical bounds.

Findings

For $L(G)= ext{\{3,3+2l\}}$, $oxed{ ext{chromatic number}= ext{max}igrace{3, ext{clique number}}ig}$.

For $L(G)= ext{\{k,k+2l\}}$, $oxed{ ext{chromatic number}=3}$.

Provides a complete solution to the problem of coloring graphs with two odd cycle lengths.

Abstract

In this paper we determine the chromatic number of graphs with two odd cycle lengths. Let $G$ be a graph and $L(G)$ be the set of all odd cycle lengths of $G$. We prove that: (1) If $L(G)=\{3,3+2l\}$, where $l\geq 2$, then $\chi(G)=\max\{3,\omega(G)\}$; (2) If $L(G)=\{k,k+2l\}$, where $k\geq 5$ and $l\geq 1$, then $\chi(G)=3$. These, together with the case $L(G)=\{3,5\}$ solved in \cite{W}, give a complete solution to the general problem addressed in \cite{W,CS,KRS}. Our results also improve a classical theorem of Gy\'{a}rf\'{a}s which asserts that $\chi(G)\le 2|L(G)|+2$ for any graph $G$.