# Uniform decompositions of complete multigraphs

We examine the notion of uniformity in the context of decompositions of complete multigraphs. A decomposition D_1,D_2,...,D_r of a graph is uniform if there exists some graph H such that D_i\cup D_j is isomorphic to H whenever 1\leq i<j\leq r. Outside of 1-factorisations, questions concerning the existence of uniform graph decompositions have not been considered previously. We prove that uniform m-cycle decompositions of the complete multigraph \mu K_n exist only when m=n, when \mu=2 and m=n-1, when \mu=1 and m=(n-1)/2, and possibly when \mu=1 and 2m(m+1)=n(n-1). If time permits, some results on uniform decompositions into graphs other than cycles may also be presented

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We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

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