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Covering and Euler cycles on non-oriented graphs

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Abstract

A covering cycle is a closed path that traverses each edge of a graph at least once. Two cycles are equivalent if one is a cyclic permutation of the other. We compute the number of equivalence classes of non-periodic covering cycles of given length in a non-oriented connected graph. A special case is the number of Euler cycles (covering cycles that cover each edge of the graph exactly once) in the non-oriented graph. We obtain an identity relating the numbers of covering cycles of any length in a graph to a product of determinants.

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Acknowledgements

G.A.T.F.C is deeply grateful to Professor M. Krebs (State University of California) for email correspondence and help with Theorem 2.1. Thanks to Prof. Asteroide Santana (UFSC) for help with latex commands and determinants, and the referee for several comments and ideas that helped to improve the paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, SC, Brasil

    G. A. T. F. da Costa & M. Policarpo

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  1. G. A. T. F. da Costa
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  2. M. Policarpo
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Correspondence to G. A. T. F. da Costa.

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da Costa, G.A.T.F., Policarpo, M. Covering and Euler cycles on non-oriented graphs. Period Math Hung 77, 201–208 (2018). https://doi.org/10.1007/s10998-018-0235-2

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