Coloring graphs with two odd cycle lengths

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 $l\ge 2$, then $\chi (G)=\max \{3,\omega (G)\}$; (2) If $L(G)=\{k,k+2l\}$, where $k\ge 5$ and $l\ge 1$, 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 $\chi (G)\le 2\mathrm{\mid }L(G)\mathrm{\mid }+2$ for any graph $G$.