The distinguishing chromatic number χD(G) of a graph G is the minimum number of colours required to properly colour the vertices of G so that the only automorphism of G that preserves colours is the identity. For a graph G of order n, it is clear that 1 6 χD(G) 6 n, and it has been shown that χD(G) = n if and only if G is a complete multipartite graph. This paper characterizes the graphs G of o...

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius’s method, which is also a modification of Olah’s method for encoding the spanning trees of any complete multipartite graph K(n1, . . . , nr). We also give a bijection between the spanning trees of a planar graph and those of any of ...

An r-edge-coloring of a graph G is a surjective assignment of r colors to the edges of G. The monochromatic tree partition number of an redge-colored graph G is defined to be the minimum positive integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In this paper, we give a direct proof for the m...

We prove that the necessary divisibility conditions are sufficient for the existence of resolvable group divisible designs with a fixed number of sufficiently large groups. Our method combines an application of the Rees product construction with a streamlined recursion based on incomplete transversal designs. With similar techniques, we also obtain new results on decompositions of complete mult...

For every directed graph D we consider the complex of all directed subforests Δ(D). The investigation of these complexes was started by D. Kozlov. We generalize a result of Kozlov and prove that complexes of directed trees of complete multipartite graphs are shellable. We determine the h-vector of Δ(Km,n) and the homotopy type of Δ( −→ Kn1,n2,...,nk ).

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to multipartite structure, separated by a phase transition from a region of disordered graphs. 1 Department of Mathematics University of Minnesota Minneapolis, MN 55455 2 Department of Mathematics University of Texas Austin, TX 78712 1

The complete multipartite graph Kn(m) with n parts of size m is shown to have a decomposition into n-cycles in such a way that each cycle meets each part of Kn(m); that is, each cycle is said to be gregarious. Furthermore, gregarious decompositions are given which are also resolvable. © 2007 Elsevier B.V. All rights reserved.

Given a finite group G of even order, which graphs Γ have a 1−factorization admitting G as automorphism group with a sharply transitive action on the vertex-set? Starting from this question, we prove some general results and develop an exhaustive analysis when Γ is a complete multipartite graph and G is cyclic.

We show that for any 2-factor U of Kn,n with n odd, there exists a hamilton decomposition of Kn,n − E(U) with a 1-factor leave. Settling this last open case now provides necessary and sufficient conditions for a hamilton decomposition of any complete multipartite graph with the edges of any 2-factor removed.