3 research outputs found
Clustering powers of sparse graphs
We prove that if is a sparse graph --- it belongs to a fixed class of
bounded expansion --- and is fixed, then the
th power of can be partitioned into cliques so that contracting each of
these clique to a single vertex again yields a sparse graph. This result has
several graph-theoretic and algorithmic consequences for powers of sparse
graphs, including bounds on their subchromatic number and efficient
approximation algorithms for the chromatic number and the clique number.Comment: 14 page
From - to -bounded classes
-bounded classes are studied here in the context of star colorings and
more generally -colorings. This leads to natural extensions of the
notion of bounded expansion class and to structural characterization of these.
In this paper we solve two conjectures related to star coloring boundedness.
One of the conjectures is disproved and in fact we determine which weakening
holds true. We give structural characterizations of (strong and weak)
-bounded classes. On the way, we generalize a result of Wood relating
the chromatic number of a graph to the star chromatic number of its
-subdivision. As an application of our characterizations, among other
things, we show that for every odd integer even hole-free graphs
contain at most holes of length .Comment: To the memory of Robin Thoma
Erd\H{o}s-Hajnal properties for powers of sparse graphs
We prove that for every nowhere dense class of graphs , positive
integer , and , the following holds: in every -vertex
graph from one can find two disjoint vertex subsets
such that and
and either for
all and , or for all and
. We also show some stronger variants of this statement, including a
generalization to the setting of First-Order interpretations of nowhere dense
graph classes