Hamiltonicity of covering graphs of trees

In this paper, we consider covering graphs obtained by lifting a tree with a loop at each vertex as a voltage graph over a cyclic group. We generalize a tool of Hell et al. (2020), known as the billiard strategy, for constructing Hamiltonian cycles in the covering graphs of paths. We show that our extended tool can be used to provide new sufficient conditions for the Hamiltonicity of covering graphs of trees that are similar to those of Batagelj and Pisanski (1982) and of Hell et al. (2020). Next, we focus specifically on covering graphs obtained from trees lifted as voltage graphs over cyclic groups $Z_p$ of large prime order $p$. We prove that for a given reflexive tree $T$ whose edge labels are assigned uniformly at random from a finite set, the corresponding lift is almost surely Hamiltonian for a large enough prime-ordered cyclic group $Z_p$. Finally, we show that if a reflexive tree $T$ is lifted over a group $Z_p$ of a large prime order, then for any assignment of nonzero elements of $Z_p$ to the edges of $T$, the corresponding cover of $T$ has a large circumference.