Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment

Abstract

We construct new resolvable decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: layering, which allows us to obtain 2‐factorizations of complete multigraphs from existing 2‐factorizations of complete graphs, and detachment, which allows us to construct resolvable cycle decompositions of complete equipartite multigraphs from existing resolvable cycle decompositions of complete multigraphs. These techniques are applied to obtain new 2‐factorizations of a specified type for both complete multigraphs and complete equipartite multigraphs, with the emphasis on new solutions to the Oberwolfach Problem and the Hamilton–Waterloo Problem. In addition, we show existence of some thickly resolvable cycle decompositions.

On the generalized Oberwolfach problem

“…For more results concerning the solvability of infinitely many instances of OP(K * v ; F ) we refer the reader to the survey [38, Section VI.12] updated to 2006. Some recent results concerning OP(λK v ; F ), when λ > 1, can be found in [8]. We point out that a solution to OP (2K 2n ; F ) -which cannot be obtained by doubling a solution of OP (2K 2n , F ) when F is not bipartite -together with a solution to OP (K 2n ± I, F ) would give rise to a solution of OP ((λK 2n )± I; F ) for every λ > 2.…”