Hamiltonicity of covering graphs of trees

TL;DR

This paper investigates Hamiltonian cycles in covering graphs of trees, extending existing tools, providing new sufficient conditions, and demonstrating that random lifts over large prime cyclic groups are almost surely Hamiltonian with large circumference.

Contribution

It extends the billiard strategy for Hamiltonicity, introduces new conditions for covering graphs of trees, and proves probabilistic Hamiltonicity results for lifts over large prime cyclic groups.

Findings

Extended billiard strategy for Hamiltonian cycles.

New sufficient conditions for Hamiltonicity of covering graphs.

Almost sure Hamiltonicity of lifts over large prime cyclic groups.

Abstract

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, Nishiyama, and Stacho, 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 and of Hell, Nishiyama, and Stacho. Next, we focus specifically on covering graphs obtained from trees lifted as voltage graphs over cyclic groups $\mathbb Z_p$ 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 $\mathbb Z_p$. Finally, we show that if a reflexive tree $T$ is lifted over a group $\mathbb Z_p$ of a large prime order, then for any assignment of nonzero elements of $\mathbb Z_p$ to the edges of $T$, the corresponding cover of $T$ has a large circumference.