The Hamilton-Waterloo Problem with $C_4$ and $C_m$ Factors

TL;DR

This paper investigates the Hamilton-Waterloo problem involving decompositions of complete graphs into 4-cycles and m-cycles, providing a comprehensive classification of solutions for odd m ≥ 3 with few exceptions.

Contribution

It fully characterizes solutions to the Hamilton-Waterloo problem with 4-cycle and m-cycle factors for odd m ≥ 3, extending previous results and resolving many cases.

Findings

Complete solutions for odd m ≥ 3 with few exceptions.

Determination of all possible 4-cycle and m-cycle factorizations.

Extension of Hamilton-Waterloo problem classifications.

Abstract

The Hamilton-Waterloo problem with uniform cycle sizes asks for a $2-$ factorization of the complete graph $K_v$ (for odd {\em v}) or $K_v$ minus a $1-$factor (for even {\em v}) where $r$ of the factors consist of $n-$cycles and $s$ of the factors consist of $m-$cycles with $r+s=\left \lfloor \frac{v-1}{2} \right \rfloor$. In this paper, the Hamilton-Waterloo Problem with $4-$cycle and $m-$cycle factors for odd $m\geq 3$ is studied and all possible solutions with a few possible exceptions are determined.