Graphs with girth $2\ell+1$ and without longer odd holes are $3$-colorable

Since $C_{1},C_{2}$ are odd holes and $g(G)=2\ell+1$, we have $|P_{1}|\leq\ell$. Assume that $1\notin\{|Q_{1}|,|Q_{2}|,|L_{1}|,|L_{2}|\}$. Since $P_{2}Q_{2}$ is a chordal path of $C_{1}$ and $C_{4}$ is an even hole, we have $|P_{2}|+|Q_{2}|=|L_{1}|$ by Lemma 2.3. Similarly, we have $|P_{2}|+|L_{1}|=|Q_{2}|$, which is a contradiction. So (1) holds.

Now we consider (2). By (1) and symmetry we may assume that $|Q_{1}|=1$. Since $|P_{1}|\leq\ell$ by (1), we have $|L_{1}|\geq\ell$, so $Q_{1}P_{2}$ is a chordal path of $C_{2}$. Then $|Q_{1}P_{2}|\geq\ell+1$ by Lemma 2.3. So $|P_{2}|\geq\ell$. This proves (2).

Next, we prove that (3) is true. Assume not. Let $v\in V(G)-V(H)$ have at least two neighbours in $V(H)$. Since $|L_{2}|>1$ and $C_{3}$ is an even hole, $C_{2}\Delta C_{3}$ is an odd hole and $|L_{2}|\geq\ell+1$ by Lemma 2.3. Moreover, since a vertex not in an odd hole has at most one neighbour in the odd hole, each arris of $H$ has at most one vertex adjacent to $v$.

Note that $C_{2}\Delta C_{3}$ is an odd hole. Since $v$ can not have two neighbours in an odd hole of $H$, the vertex $v$ has no neighbour in $V(P_{1})$. For the same reason, if $v$ has a neighbour in $V(Q_{1}\cup Q_{2})$, the neighbours of $v$ in $V(H)$ must be in $V(C_{1}\cup C_{2})$, which is not possible by 3.1.1. Hence, $N(v)\cap V(C_{1}\cup C_{2})\subseteq V(L^{*}_{1}\cup L^{*}_{2})$. Assume that $v$ has a neighbour in $V(L^{*}_{1})$. Since $|C_{4}|\leq 4\ell-2$ and $g(G)=2\ell+1$, by 3.1.1, the two cycles in $G[V(C_{4})\cup\{v\}]$ containing $v$ are odd holes. Moreover, since $|L_{2}|\geq\ell+1$, the subgraph $G[V(C_{4})\cup\{v\}]\cup P_{1}\cup Q_{1}$ is an odd $K_{4}$-subdivision, which is not possible. Similarly we can show that $v$ can not have a neighbour in $V(L^{*}_{2})$. This proves (3). ∎

By Theorem 1.2, $G$ does not have an odd $K_{4}$-subdivision. Assume to the contrary that $G$ has a balanced $K_{4}$-subdivision $H$ of type $(1,2)$ that is pictured as the graph in Figure 1. Without loss of generality we may assume that $H$ is chosen with $|H|$ as small as possible.By Lemma 3.1 (1) and symmetry we may assume that $|Q_{1}|=1$. By Lemmas 3.1 (2) and 2.3, we have

Let $e,f$ be the edges in $P_{2}$ incident with $u_{3},u_{4}$, respectively. Let $P$ be a direct connection in $G\backslash\{e,f\}$ linking $P^{*}_{2}$ and $H-V(P^{*}_{2})$. Lemma 2.1 implies that such $P$ exists. Let $v_{1},v_{2}$ be the ends of $P$ with $v_{2}$ having a neighbour in $V(P^{*}_{2})$ and $v_{1}$ having a neighbour in $V(H)-V(P^{*}_{2})$. By Lemma 3.1 (3), both $v_{1}$ and $v_{2}$ have a unique neighbour in $V(H)$. Let $x$ be the neighbour of $v_{1}$ in $V(H)$ and $y$ the neighbour of $v_{2}$ in $V(H)$. Set $P^{\prime}:=xv_{1}Pv_{2}y$. Then $H\cup P^{\prime}$ is an induced subgraph of $G$. By Lemma 3.1 (3) again, we have $|P^{\prime}|\geq 3$.

Let $u^{\prime}_{4}$ be the neighbour of $u_{4}$ in $L_{2}$.

Let $v$ be a vertex in $V(P^{*}_{2})$. When $|P_{2}|\geq\ell+1$, let $|P_{2}(u_{3},v)|=\ell-1$, otherwise let $|P_{2}(u_{3},v)|=\ell-2\geq 3$. The vertex $v$ is well defined as $|P_{2}|\geq\ell$. Since $|P_{2}|\leq 2\ell-2$, we have $2\leq|P_{2}(u_{4},v)|\leq\ell-1$. Let $u,w\in N(v)\cap V(P^{*}_{2})$ with $u$ closer to $u_{4}$ and $w$ closer to $u_{3}$ along $P_{2}$. Since $\{uv,vw\}$ is not an edge cut of $G$ by Lemma 2.1, there is a shortest induced path $Q$ in $G\backslash\{uv,vw\}$ linking $v$ and $H-v$. By Lemma 3.1 (3), $H\cup Q$ is an induced subgraph of $G$. Let $v^{\prime}$ be the other end of $Q$ that is not equal to $v$. When $v^{\prime}\in V(C_{1}\cup C_{2})$, since $Q^{*}$ is a direct connection in $G\backslash\{e,f\}$ linking $P_{2}^{*}$ and $H-V(P^{*}_{2})$, we get a contradiction to 3.2.2-3.2.6. So $v^{\prime}\in V(P^{*}_{2})-\{v\}$. Assume that $v^{\prime}\in V(P_{2}^{*}(u_{3},v))$. When $v^{\prime}\neq w$, either $Q_{2}P_{1}Q_{1}P_{2}(u_{3},v^{\prime})QP_{2}(v,u_{4})$ is an odd hole of length at least $2\ell+3$ or $QP_{2}(v,v^{\prime})$ is a cycle of length $2|P_{2}(v,v^{\prime})|$ that is at most $2\ell-2$ by the choice of $v$, which is not possible. So $v^{\prime}=w$. By symmetry, we have $v^{\prime}\in\{u,w\}$.

Now we show that $\{v,v^{\prime}\}$ is a $K_{2}$-cut of $G$, implying that Theorem 3.2 holds from Lemma 2.2. Assume not. Let $R^{\prime}$ be a direct connection in $G-\{v,v^{\prime}\}$ linking $Q^{*}$ and $H-\{v,v^{\prime}\}$. Let $s_{1},s_{2}$ be the ends of $R^{\prime}$ with $s_{1}$ having a neighbour in $V(H)-\{v,v^{\prime}\}$ and $s_{2}$ having a neighbour in $V(Q^{*})$. By Lemma 3.1 (3), $s_{1}$ has a unique neighbour, say $t_{1}$, in $V(H)-\{v,v^{\prime}\}$. Since $Q$ is chosen with $|Q|$ as small as possible, $s_{2}$ has a unique neighbour, say $t_{2}$, in $V(Q^{*})$. Set $R:=t_{1}s_{1}R^{\prime}s_{2}t_{2}$. Then $H\cup Q\cup R$ is an induced subgraph of $G$ or some vertex in $\{v,v^{\prime}\}$ has a neighbour in $V(R^{\prime})$. When $t_{1}\in V(P_{2})$, let $P^{\prime}_{2}$ be an induced $(u_{3},u_{4})$-path in $P_{2}\cup Q\cup R$ that is not equal to $P_{2}$. Since $Q_{1}P^{\prime}_{2}$ is a chordal path of $C_{2}$ with length $|L_{2}|$ by Lemma 2.3, there is a cycle in $P_{2}\cup P^{\prime}_{2}$ with length at most $2\ell$ by the choice of $v$, which is not possible. So $t_{1}\in V(H)-V(P_{2})$. Then $R\cup Q^{*}$ contains a direct connection in $G\backslash\{e,f\}$ linking $P_{2}^{*}$ and $H-V(P_{2}^{*})$. Since $|P_{2}(u_{3},v)|\geq\ell-2\geq 3$ and $2\leq|P_{2}(u_{4},v)|\leq\ell-1$, by 3.2.2-3.2.6, we have that $v^{\prime}=u$, $t_{1}=u^{\prime}_{4}$, and $uu_{4}\in E(P_{2})$. Let $P^{\prime}_{2}$ be an induced $(u_{3},u^{\prime}_{4})$-path in $(P_{2}-\{u_{4}\})\cup Q\cup R$. Since $Q_{1}P^{\prime}_{2}$ and $Q_{1}P_{2}$ are chordal paths of $C_{2}$, by Lemma 2.3, we have $|P^{\prime}_{2}(u^{\prime}_{4},v)|=|P_{2}(u_{4},v)|-1$, so $u^{\prime}_{4}v\in E(G)$, which is not possible. ∎

When $P$ is short, the lemma holds from its definition. So we may assume that $P$ is local. Since some vertex in $V(Q^{*}_{1})$ has a neighbour in $V(P^{*})$ and $g(G)=2\ell+1$, we have $|P|\geq 2\ell$. So $PQ_{2}$ is an even hole. This proves Lemma 4.1. ∎

Without loss of generality we may assume that $P$ is not a short jump over $C$. Let $s,t$ be the ends of $P$ and $Q_{1},Q_{2}$ be the internally disjoint $(s,t)$-paths of $C$. Since $P$ is neither local nor short, there are $u_{1},u_{2}\in V(P^{*})$ such that $u_{1}$ has a neighbour in $V(Q_{1}^{*})$ and $u_{2}$ has a neighbour in $V(Q_{2}^{*})$, and such that no vertex in $V(P^{*}(u_{1},u_{2}))$ has a neighbour in $V(C)$. Since no vertex outside an odd hole has two neighbours in the odd hole, $u_{1}\neq u_{2}$ and $u_{i}$ has a unique neighbour, say $w_{i}$, in $V(C)$ for each integer $1\leq i\leq 2$. Then $w_{1}u_{1}P(u_{1},u_{2})u_{2}w_{2}$ is a short jump over $C$. This proves Lemma 4.2. ∎

Assume not. Among all local jumps over $C$ not satisfying Lemma 4.3, let $P$ be chosen with $|P|$ as small as possible. Let $u_{1},u_{2}\in V(P^{*})$ have a neighbour in $V(C)-\{v_{1},v_{2}\}$ that are closest to $v_{1},v_{2}$, respectively. Let $w_{1},w_{2}$ be the unique neighbour of $u_{1},u_{2}$ in $V(C)$, respectively. Since $w_{1}u_{1}P(u_{1},v_{1})$ is a short jump over $C$ when $w_{1}v_{1}\notin E(G)$, we have $w_{1}v_{1}\in E(G)$. Similarly, we have $w_{2}v_{2}\in E(G)$. Hence, when $u_{1}=u_{2}$, $w_{1}=w_{2}$ and $w_{1}$ is adjacent to $v_{1},v_{2}$, which is a contradiction. So $u_{1}\neq u_{2}$. Since $P$ is not a local jump over $C$ across one vertex, we have $w_{1}\neq w_{2}$. Let $u_{3}$ be the neighbour of $w_{1}$ in $V(P^{*})$ closest to $v_{2}$. Note that $u_{3}$ maybe equal to $u_{1}$. By the choice of $u_{2}$, we have $u_{2}\in V(P(u_{3},v_{2}))$. Set $P^{\prime}:=w_{1}u_{3}P(u_{3},v_{2})$. Since $P^{\prime}$ is a local jump over $C$ with $|P^{\prime}|<|P|$, by the choice of $P$, the subgraph $G[V(C\cup P^{\prime})]$ contains a local jump over $C$ across one vertex or a short jump over $C$, so is $G[V(C\cup P)]$, which is a contradiction. This proves Lemma 4.3. ∎