SFU Discrete Math Seminar: Zhilin Ge

We consider covering graphs obtained by lifting trees as voltage graphs over cyclic groups. We generalize a tool of Hell, Nishiyama, and Stacho, known as the billiard strategy, for constructing Hamiltonian cycles in the covering graphs of paths. The presentation will 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 Zp of large prime order p. The presentation will show that for a given reflexive tree T with random nonzero voltage labels from Zp on its edges, the corresponding lift is almost surely Hamiltonian for a large enough prime-ordered cyclic group Zp. Finally, we show that if a reflexive tree T is lifted over a group Zp of a large prime order, then for any assignment of nonzero elements of Zp to the edges of T, the corresponding cover of T has a large circumference.