A Near Cyclic (m1, m2, ..., mr )-Cycle System Of Complete Multigraph

Let v, λ be positive integers, λKv denote a complete multigraph on v vertices in which each pair of distinct vertices joining with λ edges. In this article, difference method is used to introduce a new design that decomposes 4Kv into cycles, when v ≡ 2, 10(mod12). This design merging between cyclic (m1, ..., mr ) -cycle system and near-fourfactor is called a near cyclic (m1, ..., mr ) -cycle syste

Cycle Decompositions of Complete Multigraphs

It is shown that the obvious necessary conditions for the existence of a decomposition of the complete multigraph with n vertices and with lambda edges joining each pair of distinct vertices into m-cycles, or into m-cycles and a perfect matching, are also sufficient. This result follows as an easy consequence of more general results which are obtained on decompositions of complete multigraphs into cycles of varying lengths. (C) 2010 Wiley Periodicals, Inc. J Combin Designs 19: 42-69, 201