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arXiv · 2607.20376

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

Zi-Xia Song
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Abstract

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

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Zi-Xia Song. 2026-07-22. The Erd\H{o}s-Lov\'asz Tihany Conjecture holds for all even-hole-free graphs. https://arxiv.org/abs/2607.20376

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