Coloring graphs with independence number two and no odd clique immersions

Abstract

We study the chromatic number of graphs that exclude a clique as a strong odd immersion and have independence number two. Given a graph $G$ and $t\in {\mathbb{Z}}^{+}$, we prove that if $\alpha (G)\le 2$ and $G$ has no strong odd $K_t$-immersion, then $\chi (G)\le \lceil \frac{3(t-1)}{2}\rceil$.