Some Remarks on Even-Hole-Free Graphs

A vertex of a graph is bisimplicial if the set its neighbors union two cliques; quasi-line every bisimplicial. recent result Chudnovsky and Seymour asserts that non-empty even-hole-free has vertex. Both Hadwiger's conjecture Erdős-Lovász Tihany have been shown to be true for graphs, but are open graphs. In this note, we prove $G$ with $\omega(G)<\chi(G)=s+t-1$ satisfies provided $ t\geq s > \chi(G)/3$; $9$-chromatic $\omega(G)\leq 8$ $K_4\cup K_6$ minor; all $k\geq 7$, no $K_k$ minor $(2k-5)$-colorable. Our proofs rely heavily on structural