23 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
Polynomial treewidth forces a large grid-like-minor
Robertson and Seymour proved that every graph with sufficiently large
treewidth contains a large grid minor. However, the best known bound on the
treewidth that forces an grid minor is exponential in .
It is unknown whether polynomial treewidth suffices. We prove a result in this
direction. A \emph{grid-like-minor of order} in a graph is a set of
paths in whose intersection graph is bipartite and contains a
-minor. For example, the rows and columns of the
grid are a grid-like-minor of order . We prove that polynomial
treewidth forces a large grid-like-minor. In particular, every graph with
treewidth at least has a grid-like-minor of order
. As an application of this result, we prove that the cartesian product
contains a -minor whenever has treewidth at least
.Comment: v2: The bound in the main result has been improved by using the
Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte