Coverings of general digraphs

english

2025

volume 25, issue 3

covering projection   dart   general digraph   homotopy   lifting automorphisms   mixed graph   monoid action   voltage

Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)

A unified theory of covering projections of graphs and digraphs is presented as one theory by considering coverings of general digraphs, where multiple directed and undirected edges together with oriented and unoriented loops and semiedges, are allowed. It transpires that coverings of general digraphs can display certain pathological behaviour since the naturally defined projections of their underlying graphs may not be coverings in the usual topological sense. Consequently, homotopy does not always lift, although the unique walk lifting property still holds. Yet, it is still possible to grasp such coverings algebraically in terms of the action of the fundamental monoid. This action is permutational and has certain nice properties that monoid actions in general do not have. As a consequence, such projections can be studied combinatorially in terms of voltages. The problem of isomorphism and equivalence, and in particular, the problem of lifting automorphisms, is treated in depth. All known results about covering projections of graphs are simple corollaries of just three general theorems.

20.02.2026