Coloring graphs with no induced subdivision of $K_4^{+}$

Abstract

Let $K_4^{+}$ be the 5-vertex graph obtained from $K_4$, the complete graph on four vertices, by subdividing one edge precisely once (i.e. by replacing one edge by a path on three vertices). We prove that if the chromatic number of some graph $G$ is much larger than its clique number, then $G$ contains a subdivision of $K_4^{+}$ as an induced subgraph.

4 pages - v2 corrected typos