On gamma-hyperellipticity of graphs

Alexander D. Mednykh, Ilya A. Mednykh

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 gamma-hyperelliptic if it is a two fold branched covering of a genus gamma graph. The corresponding covering involution is called gamma-hyperelliptic.

The aim of the paper is to provide a few criteria for the involution tau acting on a graph X of genus g to be gamma-hyperelliptic. If tau has at least one fixed point then the first criterium states that there is a basis in the homology group H_1(X) whose elements are either invertible or split into gamma interchangeable pairs under the action of tau_*. The second criterium is given by the formula tr_{H_1(X)}, (tau_*) = 2 gamma - g. Similar results are also obtained in the case when tau acts fixed point free.


Keywords


Graph, hyperelliptic graph, homology group, Riemann--Hurwitz formula, Schreier formula.

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DOI: https://doi.org/10.26493/1855-3974.650.00d

ISSN: 1855-3974

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