Almost 2-perfect 8-cycle systems

TL;DR

This paper investigates the existence and construction of almost 2-perfect 8-cycle systems and packings in complete graphs, establishing existence results for various orders and highlighting differences from standard cycle packings.

Contribution

It proves the existence of almost 2-perfect maximum packings of $K_n$ with 8-cycles for all $n \\geq 8$ and constructs maximum 8-cycle packings that are not almost 2-perfect for all $n \\geq 10$.

Findings

Almost 2-perfect maximum packings exist for all $n \\geq 8$.

Maximum 8-cycle packings not almost 2-perfect exist for all $n \\geq 10$.

The concept generalizes 2-perfect cycle systems and explores their existence in complete graphs.

Abstract

For an $m$-cycle $C$, an inside $m$-cycle of $C$ is a cycle on the same vertex set, that is edge-disjoint from $C$. In an $m$-cycle system, $(\mathcal{X}, \mathcal{C})$, if inside $m$-cycles can be chosen -one for each cycle- to form another $m$-cycle system, then $(\mathcal{X}, \mathcal{C})$ is called an almost $2$-perfect $m$-cycle system. Almost $2$-perfect cycle systems can be considered as generalisations of $2$-perfect cycle systems. Cycle packings are generalisations of cycle systems that allow to have leaves after decomposition. In this paper, we prove that an almost $2$-perfect maximum packing of $K_n$ with $8$-cycles of order $n$ exists for each $n\geq 8$. We also construct a maximum $8$-cycle packing of order $n$ which is not almost $2$-perfect for each $n \geq 10$.