Perfect 1-Factorisations of Circulant Graphs

Abstract

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (1-regular spanning subgraphs). A perfect 1-factorisation is a 1-factorisation in which the union of every pair of distinct 1-factors is a Hamilton cycle; a 1-factorisation is uniform if the graphs resulting from the union of every pair of distinct 1-factors are isomorphic. We initiate the study of perfect 1-factorisations and uniform 1-factorisations of circulant graphs, a well-studied family of Cayley graphs, by determining whether or not such 1-factorisations exist for even order circulant graphs of small degree. For a bipartite circulant graph, we observe that having order congruent to 2 (mod 4) is a necessary condition for the existence of a perfect 1-factorisation. We prove that for bipartite 3-regular circulant graphs this necessary condition is also sufficient. Furthermore, the bipartite 3-regular circulant graphs of order 2 (mod 4) are the only 3-regular circulant graphs of order greater than 6 which admit a perfect 1-factorisation. The 4-regular case is more complex; we provide constructions of perfect 1-factorisations for many families of bipartite 4-regular circulant graphs of order 2 (mod 4); however, we also prove that there is an infinite family of such graphs no member of which admits a perfect 1-factorisation. We conjecture that a connected non-bipartite 4-regular circulant graph of order at least 8 does not admit a perfect 1-factorisation; we prove this conjecture for three distinct infinite families of non-bipartite 4-regular circulant graphs and show by computer search that the conjecture holds for small orders (at most 30). Several of our results are achieved by studying perfect 1-factorisations of a particular family of 4-regular graphs, denoted Dh, k for integers h, knwhere h g 2 and k g 3. These graphs are shown to be isomorphic to Cayley graphs when k is even and yet they are not vertex-transitive when k is odd (and hng 3). For small values of h, we make use of the structure of Dh,k to determine when it admits a perfect 1-factorisation and then use specific isomorphism results to obtain corresponding results for 4-regular circulant graphs. In this thesis the connected 3-regular circulant graphs that admit a perfect 1-factorisation are characterised and although the connected 4-regular circulant graphs that admit a perfect 1-factorisation have not yet been fully characterised, we provide many interesting theoretical results that establish the framework for such a characterisation.n