Uniform decompositions of complete multigraphs into cycles

Abstract

AbstractThe notion of uniformity, as in uniform ‐factorisations, extends naturally to graph decompositions generally. The existence of uniform decompositions of complete multigraphs into cycles is investigated and some connections with families of classical designs are established. We show that if there exists a uniform decomposition of into ‐cycles then (A) and , or (B) and , or (C) , and or (D) and . For case A, there are only a few small values of and to consider, and we exhibit all uniform decompositions up to isomorphism for each such and . In each of cases B and C, we construct examples of uniform decompositions for infinitely many values of , and we investigate the isomorphism classes of our examples for each such . We have no examples of uniform decompositions in case D, but we rule out the smallest example, namely and , and we prove that if such decompositions exist, then so do large quasiresidual designs that are not residual.