Graphs with non-disjoint odd cycles satisfies $\chi \le 5$ Given a graph $G$ with the property that any two odd cycles in the graph share a  vertex, then show that the chromatic index of the graph, $\chi$, is at most $5$.

Let $C$ be the smallest odd cycle in $G$.  Note that $C$ has no chords (i.e. edges between its vertices that aren't part of the cycle), as otherwise there would be a smaller odd cycle.  Hence we can color $C$ with $3$ colors, and it will be a valid partial coloring of G.
Now if we throw all the vertices in $C$ from $G$, the remaining graph is bipartite because any odd cycle in $G$ must share a vertex with $C$ by assumption. Now we can color all of those vertices with two new colors. We have colored $G$ with $5$ colors, so $\chi(G) \le 5$.
