22 research outputs found
Partitioning a graph into highly connected subgraphs
Given , a -proper partition of a graph is a partition
of such that each part of induces a
-connected subgraph of . We prove that if is a graph of order
such that , then has a -proper partition with at
most parts. The bounds on the number of parts and the minimum
degree are both best possible. We then prove that If is a graph of order
with minimum degree , where
, then has a -proper partition into at most
parts. This improves a result of Ferrara, Magnant and
Wenger [Conditions for Families of Disjoint -connected Subgraphs in a Graph,
Discrete Math. 313 (2013), 760--764] and both the degree condition and the
number of parts are best possible up to the constant
Consistent random vertex-orderings of graphs
Given a hereditary graph property , consider distributions of
random orderings of vertices of graphs that are preserved
under isomorphisms and under taking induced subgraphs. We show that for many
properties the only such random orderings are uniform, and give
some examples of non-uniform orderings when they exist
A version of Szemer\'edi's regularity lemma for multicolored graphs and directed graphs that is suitable for induced graphs
In this manuscript we develop a version of Szemer\'edi's regularity lemma
that is suitable for analyzing multicolorings of complete graphs and directed
graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M.
Szegedy [Combinatorica, 20(4) (2000), 451--476] who prove a similar result for
graphs.
The purpose is to extend classical results on dense hereditary properties,
such as the speed of the property or edit distance, to the above-mentioned
combinatorial objects.Comment: 11 page
Partitioning a graph into highly connected subgraphs
Abstract Given k ≥ 1, a k-proper partition of a graph G is a partition P of V (G) such that each part P of P induces a k-connected subgraph of G. We prove that if G is a graph of order n such that δ(G) ≥ √ n, then G has a 2-proper partition with at most n/δ(G) parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that if G is a graph of order n with minimum degree where c