Almost resolvable $k$-cycle systems with $k\equiv 2\pmod 4$

TL;DR

This paper proves the existence of almost resolvable $k$-cycle systems for certain parameters, advancing the understanding of cycle decompositions in combinatorial design theory.

Contribution

It establishes the existence of almost resolvable $k$-cycle systems for all sufficiently large $t$ when $k ot ot ext{equiv} 0 ext{ mod } 4$, partially solving an open problem.

Findings

Existence of almost resolvable $k$-cycle systems for $k eq 2 ext{ mod } 4$ and large $t$

Partial resolution of an open problem in combinatorial design theory

Conditions under which such systems exist for $k extgreater 6$

Abstract

In this paper, we show that if $k\geq 6$ and $k \equiv 2 \pmod 4$, then there exists an almost resolvable $k$-cycle system of order $2kt+1$ for all $t\ge 1$ except possibly for $t=2$ and $k\geq 14$. Thus we give a partial solution to an open problem posed by Lindner, Meszka, and Rosa (J. Combin. Des., vol. 17, pp.404-410, 2009).