Uniform cycle decompositions of complete multigraphs

Abstract

A decomposition of a graph G is a set {D 1 , D 2 , ..., D r } of subgraphs of G whose edge sets partition the edge set of G. A graph decomposition is uniform if the union of any two distinct subgraphs of the decomposition is isomorphic to the union of any other two distinct subgraphs, that is, D i ∪ D j ∼ = D k ∪ D l whenever 1 ≤ i < j ≤ r, 1 ≤ k < l ≤ r. This is a natural extension of the notion of uniformity of 1-factorisations (graph decompositions in which each subgraph is a 1-factor). In this project, we initiate the study of uniform decompositions of complete multigraphs into cycles, stars and paths. A complete multigraph µK n is a graph with n vertices and precisely µ edges between each pair of vertices. An m-cycle is a connected graph with m vertices in which each vertex has degree two, that is, a graph with the vertex set {v 1 , v 2 , ..., v m } and the edge set {{v 1 , v 2 }, {v 2 , v 3 }, ..., {v m−1 , v m }, {v m , v 1 }}. We show that if there exists a uniform decomposition of µK n into mcycles then (A) n = m and n ≤ 7, or (B) µ = 2 and m = n − 1, or (C) µ = 1, m = (n − 1)/2 and n ≡ 3 (mod 4) or (D) µ = 1 and 2m(m + 1) = n(n − 1). We fully characterise the complete multigraphs which admit uniform cycle decompositions in case (A). In cases (B) and (C), we construct uniform decompositions for infinitely many values of n and categorise those decompositions into isomorphism classes. In case (D) we have no examples of uniform decompositions, but we prove that the existence of such a decomposition would imply the existence of a large quasi-residual design which is not residual. We discuss the computational methods and algorithms used to examine small cases of these problems, including the proof that there is no uniform decomposition in case (A) when n = 9 or n = 15, or in case (C) when n = 15. We also present some tables of results from those algorithms, giving the uniform decompositions in case (B) when n ≤ 11. A k-star is a connected graph in which one vertex has degree k and k vertices have degree one. An m-path is a connected graph with m + 1 vertices and m edges in which two vertices have degree one and the remaining vertices have degree two, that is, a graph with vertex set {v 1 , v 2 , ..., v m , v m+1 } and edge set {{v 1 , v 2 }, {v 2 , v 3 }, ..., {v m , v m+1 }}. We show that there exists a uniform star decomposition of µK n with n > 2 precisely when µ = 2, or µ = 1 and there exists a skew Hadamard design of order n or order n−1. Finally, we prove that uniform m-path decompositions of µK n exist only when n ≤ 6, and construct all such decompositions. v Financial support