On γ-hyperellipticity of graphs

Abstract

The basic objects of research in this paper are graphs and their branched coverings. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first homology group. A graph is said to be γ-hyperelliptic if it is a two fold branched covering of a genus γ graph. The corresponding covering involution is called γ-hyperelliptic. The aim of the paper is to provide a few criteria for the involution τ acting on a graph X of genus g to be γ-hyperelliptic. If τ has at least one fixed point then the first criterium states that there is a basis in the homology group H1(X) whose elements are either invertible or split into interchangeable pairs under the action of τ∗: The second criterium is given by the formula trH1(X) (τ∗) = 2γ-g: Similar results are also obtained in the case when τ acts fixed point free.