Octahedral graph ($K_{2,2,2}$)

The octahedral graph is the graph obtained from $K_6$ by removing a perfect matching; equivalently it is $K_{2,2,2}$. Its chromatic number is three, and it contains two vertex-disjoint triangles in a graph. A complete bipartite graph with a single additional internal edge cannot contain it: every triangle in that host uses the additional edge.