The Hamilton-Waterloo Problem for Triangle-Factors and Heptagon-Factors

TL;DR

This paper solves the Hamilton-Waterloo problem for decomposing complete graphs into triangle and heptagon factors for odd order graphs, with only three exceptions at n=21.

Contribution

It provides a solution for the Hamilton-Waterloo problem involving triangle and heptagon factors for odd n, with minimal exceptions.

Findings

Solved for odd n with triangle and heptagon factors

Only three exceptions at n=21

Advances understanding of 2-factorizations in complete graphs

Abstract

Given 2-factors $R$ and $S$ of order $n$, let $r$ and $s$ be nonnegative integers with $r+s=\lfloor \frac{n-1}{2}\rfloor$, the Hamilton-Waterloo problem asks for a 2-factorization of $K_n$ if $n$ is odd, or of $K_n-I$ if $n$ is even, in which $r$ of its 2-factors are isomorphic to $R$ and the other $s$ 2-factors are isomorphic to $S$. In this paper, we solve the problem for the case of triangle-factors and heptagon-factors for odd $n$ with 3 possible exceptions when $n=21$.