Some remarks on even-hole-free graphs

TL;DR

This paper explores structural properties of even-hole-free graphs, proving new coloring bounds and minor existence results that extend understanding of their chromatic and minor-related characteristics.

Contribution

It establishes new coloring bounds for even-hole-free graphs with no large clique minors and confirms the Erdős-Lovász Tihany conjecture under specific conditions.

Findings

Every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable for $k \\ge 7$.

Certain even-hole-free graphs satisfy the Erdős-Lovász Tihany conjecture when specific chromatic and clique size conditions are met.

Every 9-chromatic graph with clique number at most 8 has a $K_4 old K_6$ minor.

Abstract

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger's conjecture and the Erd\H{o}s-Lov\'asz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all $k\ge7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable; every even-hole-free graph $G$ with $\omega(G)<\chi(G)=s+t-1$ satisfies the Erd\H{o}s-Lov\'asz Tihany conjecture provided that $ t\ge s> \chi(G)/3$. Furthermore, we prove that every $9$-chromatic graph $G$ with $\omega(G)\le 8$ has a $K_4\cup K_6$ minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs.