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

A hole is an induced cycle of length at least 4. Let $\text{l}\ge 2$ be a positive integer, let ${\mathcal{G}}_l$ denote the family of graphs which have girth $2\text{l}+1$ and have no holes of odd length at least $2\text{l}+3$, and let $G\in {\mathcal{G}}_{\text{l}}$. For a vertex $u\in V(G)$ and a nonempty set $S\subseteq V(G)$, let $d(u, S)=\min{d(u, v):v\in S}$,andlet$L_i(S)={u\in V(G) \mbox{ and } d(u, S)=i}$foranyinteger$i\ge 0$\mathrm{.}Weshowthatif$G[S]$isconnectedand$G[L_i(S)]$isbipartiteforeach$i\in{1, \ldots, \lfloor{\l\over 2}\rfloor}$,then$G[L_i(S)]$isbipartiteforeach$i>0$,andconsequently$\chi(G)\le 4$,where$G[S]$denotesthesubgraphinducedby$S$\mathrm{.}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 graph $G\in {\mathcal{G}}_2$, 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 ${\mathcal{G}}_2$ that induces neither $\theta$ nor ${\theta }^{-}$, and minimal non-3-colorable graphs of ${\mathcal{G}}_2$ induce no ${\theta }^{+}$.