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- W3167276073 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 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 $ tgeq s > chi(G)/3$; every $9$-chromatic graph $G$ with $omega(G)leq 8$ has a $K_4cup K_6$ minor; for all $kgeq 7$, 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." @default.
- W3167276073 created "2021-06-22" @default.
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- W3167276073 date "2022-07-29" @default.
- W3167276073 modified "2023-09-26" @default.
- W3167276073 title "Some Remarks on Even-Hole-Free Graphs" @default.
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- W3167276073 doi "https://doi.org/10.37236/10772" @default.
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