The Hamilton-Waterloo Problem for $C_3$-factors and $C_n$-factors

TL;DR

This paper completely solves the Hamilton-Waterloo problem for decomposing complete graphs into 3-cycles and cycles of lengths 4, 5, or 7, advancing understanding of graph factorizations.

Contribution

The paper provides a complete solution to the Hamilton-Waterloo problem for specific cycle lengths, filling a gap in graph factorization theory.

Findings

Solved the problem for $C_3$-factors and $C_n$-factors with n=4,5,7

Established existence conditions for these factorizations

Extended previous partial results in the field

Abstract

The Hamilton-Waterloo problem asks for a 2-factorization of $K_v$ (for $v$ odd) or $K_v$ minus a $1$-factor (for $v$ even) into $C_m$-factors and $C_n$-factors. We completely solve the Hamilton-Waterloo problem in the case of $C_3$-factors and $C_n$-factors for $n=4,5,7$.