Decomposing complete equipartite multigraphs into cycles of variable lengths: the amalgamation-detachment approach

TL;DR

This paper introduces an amalgamation-detachment technique to decompose complete equipartite multigraphs into cycles of variable lengths, extending known cycle decompositions from smaller to larger graphs.

Contribution

It provides a new method for decomposing complete equipartite multigraphs into cycles, generalizing previous cycle decomposition results using the amalgamation-detachment approach.

Findings

Decomposition of $ ext{lambda} K_{n imes m}$ into cycles of lengths $c_1m,\dots,c_km$ established.

Conditions for related cycle decompositions of complete equipartite multigraphs provided.

Extension of cycle decomposition techniques to larger and more complex multigraphs achieved.

Abstract

Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph $\lambda K_{n\times m}$ can be decomposed into cycles of lengths $c_1m,\dots,c_km$ (plus a 1-factor if the degree is odd) whenever there exists a decomposition of $\lambda m K_n$ into cycles of lengths $c_1, \dots,c_k$ (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph $\lambda K_{n\times m}$.