5 research outputs found
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
Approximate packing of independent transversals in locally sparse graphs
Consider a multipartite graph with maximum degree at most , parts
have size , and every vertex has at most
neighbors in any part . Loh and Sudakov proved that any such has an
independent transversal. They further conjectured that the vertex set of
can be decomposed into pairwise disjoint independent transversals. In the
present paper, we resolve this conjecture approximately by showing that
contains pairwise disjoint independent transversals. As applications,
we give approximate answers to questions of Yuster, and of Fischer, K\"uhn, and
Osthus