Hamiltonicity of covering graphs of trees

Abstract

In this paper, we consider covering graphs obtained by lifting trees as voltage graphs over cyclic groups. We generalize a tool of Hell, Nishiyama, and Stacho [6], 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 Posanski [3] and of Hell, Nishiyama, and Stacho [6]. 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 with random nonzero voltage labels from Z p on its edges, the corresponding lift is almost surely Hamiltonian for a large enough prime-orderd 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.