Resolvable Cycle Decompositions of Complete Multigraphs and Complete Equipartite Multigraphs via Layering and Detachment

TL;DR

This paper introduces new methods called layering and detachment to create resolvable cycle decompositions of complete multigraphs and equipartite multigraphs, addressing classical problems like the Oberwolfach and Hamilton-Waterloo problems.

Contribution

It develops two novel techniques for constructing resolvable cycle decompositions, expanding solutions to longstanding combinatorial design problems.

Findings

New 2-factorizations for complete multigraphs and equipartite multigraphs.

Solutions to the Oberwolfach and Hamilton-Waterloo problems.

Existence results for certain $eta$-resolvable cycle decompositions.

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: {\em layering}, which allows us to obtain 2-factorizations of complete multigraphs from existing 2-factorizations of complete graphs, and {\em 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 $\alpha$-resolvable cycle decompositions.