Coverings of Graphoids: Existence Theorem and Decomposition Theorems

Author:

Malnič Aleksander12,Zgrablić Boris3

Affiliation:

1. Department of Mathematics and Computer Science, Faculty of Education, University of Ljubljana, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia

2. Department of Mathematics, IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia

3. Department of Mathematics, FAMNIT, Univerza na Primorskem, Glagoljaška 8, 6000 Koper, Slovenia

Abstract

A graphoid is a mixed multigraph with multiple directed and/or undirected edges, loops, and semiedges. A covering projection of graphoids is an onto mapping between two graphoids such that at each vertex, the mapping restricts to a local bijection on incoming edges and outgoing edges. Naturally, as it appears, this definition displays unusual behaviour since the projection of the corresponding underlying graphs is not necessarily a graph covering. Yet, it is still possible to grasp such coverings algebraically in terms of the action of the fundamental monoid and combinatorially in terms of voltage assignments on arcs. In the present paper, the existence theorem is formulated and proved in terms of the action of the fundamental monoid. A more conventional formulation in terms of the weak fundamental group is possible because the action of the fundamental monoid is permutational. The standard formulation in terms of the fundamental group holds for a restricted class of coverings, called homogeneous. Further, the existence of the universal covering and the problems related to decomposing regular coverings via regular coverings are studied in detail. It is shown that with mild adjustments in the formulation, all the analogous theorems that hold in the context of graphs are still valid in this wider setting.

Funder

Ministrstvo za visoko šolstvo, znanost in tehnologijo Slovenije

Publisher

MDPI AG

Reference31 articles.

1. Malnič, A., and Zgrablić, B. (2023). Covers of general digraphs. Ars Math. Contemp., submitted for publication.

2. Biggs, N.L. (1974). Algebraic Graph Theory, Cambridge University Press.

3. Automorphisms of graphs and coverings;J. Comb. Theory Ser. B,1974

4. Voltage graphs;Gross;Discret. Math.,1974

5. Observations on the construction of covers using permutation voltage assignments;Ezell;Discret. Math.,1979

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