The Erd\H{o}s-Lov\'asz Tihany Conjecture holds for all even-hole-free graphs

Let $s, t\ge2 $ be integers.

A graph $G$ is $(s,t)$-splittable if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $\chi(G[S ]) \ge s$ and $\chi(G[T ]) \ge t$.

The Erdős-Lovász Tihany Conjecture from 1968 asserts that every graph $G$ satisfying $\omega(G)<\chi(G)=s+t-1$ is $(s,t)$-splittable.

A vertex of a graph is bisimplicial if the set of its neighbors can be expressed as the union of two cliques.

Let $G$ be a graph with $\omega(G)<\chi(G)=s+t-1$.

We prove that if $G$ does not contain $C_4$ as an induced subgraph and every induced subgraph of $G$ has a bisimplicial vertex, then $G$ is $(s,t)$-splittable.

Combining our result with a recent result of Chudnovsky and Seymour, which states that every non-empty even-hole-free graph has a bisimplicial vertex, we obtain that the Erdős-Lovász Tihany Conjecture holds for all even-hole-free graphs.