On coloring of graphs of girth $2\l+1$ without longer odd holes

Abstract

A hole is an induced cycle of length at least 4. Let $\l\ge~2$ be an integer and $r=\min\{\l-1,~\lfloor{l\over~2}\rfloor+1\}$, let ${\cal~G}_l$ denote the family of graphs which have girth $2\l+1$ and have no holes of odd length at least $2\l+3$, and let $G\in~{\cal~G}_{\l}$. For a vertex $u\in~V(G)$ and a nonempty set $S\subseteq~V(G)$, let $d(u,~S)=\min\{d(u,~v):v\in~S\}$, and let $L_i(S)=\{u\in~V(G)~\mbox{~and~}d(u,~S)=i\}$ for any integer $i\ge~0$. We show that if $G[S]$ is connected and $G[L_i(S)]$ is bipartite for each $i\in\{1,~\ldots,~r\}$, then $G[L_i(S)]$ is bipartite for each $i>0$, and consequently $\chi(G)\le~4$, where $G[S]$ denotes the subgraph induced by $S$.Let $\theta^-$ be the graph obtained from the Petersen graph by deleting three vertices which induce a path, let $\theta^+$ be the graph obtained from the Petersen graph by deleting two adjacent vertices, and let $\theta$ be the graph obtained from $\theta^+$ by removing an edge incident with two vertices of degree 3. For a non-Petersen graph $G\in{\cal~G}_2$, we show that if $G$ is 3-connected and has no unstable 3-cutset then $G$ must induce either $\theta$ or $\theta^-$ but does not induce $\theta^+$. As corollaries, $\chi(G)\le~3$ for every graph $G$ of ${\cal~G}_2$ that induces neither $\theta$ nor $\theta^-$, and minimal non-3-colorable graphs of ${\cal~G}_2$ induce no $\theta^+$.