392 research outputs found
Hitting all maximum cliques with a stable set using lopsided independent transversals
Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a
stable set meeting all maximum cliques. We strengthen this result, proving that
such a stable set exists for any graph with omega > (2/3)(Delta+1). This is
tight, i.e. the inequality in the statement must be strict. The proof relies on
finding an independent transversal in a graph partitioned into vertex sets of
unequal size.Comment: 7 pages. v4: Correction to statement of Lemma 8 and clarified proof
A different short proof of Brooks' theorem
Lov\'asz gave a short proof of Brooks' theorem by coloring greedily in a good
order. We give a different short proof by reducing to the cubic case. Then we
show how to extend the result to (online) list coloring via the Kernel Lemma.Comment: added cute Kernel Lemma trick to lift up to (online) list colorin
A note on coloring vertex-transitive graphs
We prove bounds on the chromatic number of a vertex-transitive graph
in terms of its clique number and maximum degree . We
conjecture that every vertex-transitive graph satisfies and we
prove results supporting this conjecture. Finally, for vertex-transitive graphs
with we prove the Borodin-Kostochka conjecture, i.e.,
- β¦