Intersecting Odd Cycles, Chromatic Number, and the Subgraph $K_5$

Let me give few ideas that might help, although I do not have a complete proof (yet).

Suppose that $G$ has intersecting odd cycles and that $G$ has no $K_5$ as subgraph. We shall prove that $\chi(G)\leq 4$.

Let us suppose that $G$ contains no triangle. Let $C$ be the smallest odd cycle of $G$ (which has at least five vertices). $G-C$ is bipartite with parts $A$ and $B$. Note that every vertex of $A\cup B$ has at most two neighbors in $C$, because $G$ has no triangles and $C$ is a minimum odd cycle of $G$ (having at least three neighbors in $C$, imply on two neighbors at distance at least four in $C$, since $G$ has no triangles, and thus on a smaller odd cycle). Consequently, one can extend a coloring of $C$ with colors 1,2 and 3, to a coloring of $G$ by coloring the whole set $A$ with color 4 and coloring each vertex of $B$ with a color in the set ${1,2,3}$ that does not appear in its neighborhood in $C$.

The problem is then how to solve the case when $G$ has a triangle $C$? I do not know yet how to extend a 3-coloring of the triangle to a 4-coloring of $G$. Of course, the hypothesis of having intersecting odd cycles and having no $K_5$ as subgraph are important now.

For instance, one can prove that if $G-C$ has parts $A$ and $B$ and $A_i$ (resp. $B_i$) corresponds to the set of vertices in $A$ (resp. $B$) with exactly $i$ neighbors in $C$, for every $i\in\{0,\ldots,3\}$, then $A_3\neq \emptyset$ and $B_3\neq\emptyset$, as otherwise it is possible to find a 4-coloring of $G$ by a similar argument as in the previous case. Supposing that $A_3$ and $B_3$ are both non-empty lead us to deduce that $A_3\cup A_2\cup B_3 \cup B_2$ is an independent set, as otherwise one may find a $K_5$ or two disjoint triangles. Since we do not have two disjoint triangles, $A_3\cup B_1$ and $B_3\cup A_1$ are also independent sets. Moreover, there is no edge $a_1b_1$ such that $a_1\in A_1$, $b_1\in B_1$ and $a_1$ and $b_1$ have the same neighbor in $C$.

Yet, I cannot color $G$. The vertices with no neighbor in $C$ are an issue.