Cyclic hamiltonian cycle systems of the complete multipartite graph: even number of parts

TL;DR

This paper provides a complete characterization of the existence of cyclic Hamiltonian cycle systems in complete multipartite graphs with an even number of parts, extending previous results for complete graphs and their variants.

Contribution

It offers a full solution to the existence problem of cyclic HCS in $K_{m\times n}$ for even $m$, and establishes necessary and sufficient conditions for cyclic and symmetric HCS.

Findings

Complete solution for cyclic HCS in $K_{m\times n}$ with even $m$.

Necessary and sufficient conditions for cyclic and symmetric HCS.

Extends prior work on Hamiltonian cycle systems in complete graphs.

Abstract

A hamiltonian cycle system (HCS, for short) of a graph $\Gamma$ is a partition of the edges of $\Gamma$ into hamiltonian cycles. A HCS is cyclic when it is invariant under a cyclic permutation of all the vertices of $\Gamma$; the existence problem for a cyclic HCS has been completely solved by Buratti and Del Fra in 2004 when $\Gamma$ is the complete graph $K_v$, $v$ odd, and by Jordon and Morris in 2008 when $\Gamma$ is the complete graph minus a $1$-factor $K_v-I$, $v$ even. In this work we present a complete solution to the existence problem of a cyclic HCS for $\Gamma = K_{m\times n}$, the complete multipartite graph, when the number of parts $m$ is even. We also give necessary and sufficient conditions for the existence of a cyclic and symmetric HCS of $\Gamma$; the notion of a symmetric HCS of a graph $\Gamma$ has been introduced in 2004 by Akiyama, Kobayashi, and Nakamura for $\Gamma =K_v$, $v$ odd, in 2011 by Brualdi and Schroeder when $\Gamma = K_v-I$, $v$ even, and, very recently, by Schroeder when $\Gamma$ is the complete multipartite graph.