# 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

### About Pure mathematics seminars

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.

Seminars are held on Tuesdays from 3pm to 3.50pm in Room 67-442 of the Priestley Building (Building 67).

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

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#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar.