Regular packings of regular graphs

Abstract

A graph H is G-decomposable if it contains subgraphs G 1,…,G h, h⩾2 , isomorphic to G whose sets of edges partition E( H). Wilson (Proceedings of the Fifth British Combinatorial Conference, University of Aberdeen, Aberdeen, 1975, pp. 647–659; Congr. Numer. XV, Utilitas, Math., Winnipeg, Manitoba, 1976) proved that, given a nonempty graph G, the complete graph K N is G-decomposable for N large enough, provided some natural divisibility conditions hold. Fink and Ruiz (Czechoslovak Math. J. 36 (111) (1986) 172) proved that a noncomplete G-decomposable graph H exists even within the class of circulant graphs. The order N 0( G) of the smallest G-decomposable regular graph is known only for particular classes of graphs or for graphs with small maximum degree. We give some tools to study the problem of determining N 0( G) when G is a connected regular graph. These tools are applied to obtain upper and lower bounds of N 0( G) for regular graphs of degree r⩾| V( G)|/2. Families of extremal graphs which attain the bounds are also given.