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 $G$.