Minimum H-decompositions of graphs: Edge-critical case

Abstract

For a given graph H let phi(H)(n) be the maximum number of parts that are needed to partition the edge set of any graph on n vertices such that every member of the partition is either a single edge or it is isomorphic to H. Pikhurko and Sousa conjectured that phi(H)(n) = ex(n, H) for chi (H) >= 3 and all sufficiently large n, where ex(n, H) denotes the maximum size of a graph on n vertices not containing H as a subgraph. In this article, their conjecture is verified for all edge-critical graphs. Furthermore, it is shown that the graphs maximizing phi(H) (n) are (chi(H) - 1)-partite Turan graphs. (C) 2011 Elsevier Inc. All rights reserved.