Some Remarks on Even-Hole-Free Graphs
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 $\omega(G)<\chi(G)=s+t-1$ satisfies the Erdős-Lovász Tihany conjecture provided that $ t\geq s > \chi(G)/3$; every $9$-chromatic graph $G$ with $\omega(G)\leq 8$ has a $K_4\cup K_6$ minor; for all $k\geq 7$, 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.