Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

TL;DR

This paper proves a conjecture about the existence of cyclic two-factorizations of complete graphs with specific cycle types for infinitely many cases, advancing understanding in combinatorial design theory.

Contribution

The authors confirm the conjecture for cases where \\ell \\equiv 1 \\pmod{4} and m \\geq \\ell^2 - \\ell + 1, establishing infinite families of solutions.

Findings

Confirmed the conjecture for \\ell \\equiv 1 \\pmod{4} and large enough m.

Established existence of infinitely many cyclic solutions.

Extended the class of known solutions to the Hamilton-Waterloo problem.

Abstract

It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of type $m^{\ell}$. The authors prove the conjecture in the affirmative when $\ell \equiv 1\; \pmod{4}$ and $m \geq \ell^2 -\ell + 1$.