A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs

TL;DR

This paper extends the Hamilton-Waterloo problem to complete equipartite graphs, providing new decomposition methods for these graphs into 2-factors with cycles of specified lengths, under certain conditions.

Contribution

It generalizes the Hamilton-Waterloo problem to complete equipartite graphs and introduces new decomposition constructions for 2-factors with cycles of specified lengths.

Findings

Decomposition of $K_{(xyzw:m)}$ into specified 2-factors is possible under certain gcd and modular conditions.

New constructions allow cycles of different lengths within the same 2-factor.

Results solve instances of the Hamilton-Waterloo problem for complete graphs using these generalized methods.

Abstract

The Hamilton-Waterloo problem asks for which $s$ and $r$ the complete graph $K_n$ can be decomposed into $s$ copies of a given 2-factor $F_1$ and $r$ copies of a given 2-factor $F_2$ (and one copy of a 1-factor if $n$ is even). In this paper we generalize the problem to complete equipartite graphs $K_{(n:m)}$ and show that $K_{(xyzw:m)}$ can be decomposed into $s$ copies of a 2-factor consisting of cycles of length $xzm$; and $r$ copies of a 2-factor consisting of cycles of length $yzm$, whenever $m$ is odd, $s,r\neq 1$, $\gcd(x,z)=\gcd(y,z)=1$ and $xyz\neq 0 \pmod 4$. We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton-Waterloo problem for complete graphs.