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

A hamiltonian cycle system (HCS, for short) of a graph $Γ$ is a partition of the edges of $Γ$ into hamiltonian cycles. A HCS is cyclic when it is invariant under a cyclic permutation of all the vertices of $Γ$; the existence problem for a cyclic HCS has been completely solved by Buratti and Del Fra in 2004 when $Γ$ is the complete graph $K_v$, $v$ odd, and by Jordon and Morris in 2008 when $Γ$ 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 $Γ= K_{m\times n}$, 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 $Γ$; the notion of a symmetric HCS of a graph $Γ$ has been introduced in 2004 by Akiyama, Kobayashi, and Nakamura for $Γ=K_v$, $v$ odd, in 2011 by Brualdi and Schroeder when $Γ= K_v-I$, $v$ even, and, very recently, by Schroeder when $Γ$ is the complete multipartite graph.