Some remarks on even-hole-free graphs
ZX Song - arXiv preprint arXiv:2106.01136, 2021 - arxiv.org
arXiv preprint arXiv:2106.01136, 2021•arxiv.org
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) …
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) …
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 , every even-hole-free graph with no minor is -colorable; every even-hole-free graph with satisfies the Erd\H{o}s-Lov\'asz Tihany conjecture provided that . Furthermore, we prove that every -chromatic graph with has a minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs.
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