Some Remarks on Even-Hole-Free Graphs - Song, Zi-Xia (The Electronic Journal of Combinatorics, 2022)
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 Erdős-Lovász 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 every even-hole-free graph G with ω(G)<χ(G)=s+t-1 satisfies the Erdős-Lovász Tihany conjecture provided that t≥ s > χ(G)/3; every 9-chromatic graph G with ω(G)≤ 8 has a K_4K_6 minor; for all k≥ 7, every even-hole-free graph with no K_k minor is (2k-5)-colorable. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs.