1. Coverings of general digraphsAleksander Malnič, Boris Zgrablić, 2025, original scientific article Abstract: 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.
Keywords: mixed graph, general digraph, dart, covering projection, voltage, homotopy, monoid action, lifting automorphisms
Published in RUP: 10.09.2025; Views: 486; Downloads: 17
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2. On reduced Hamilton walksAleksander Malnič, Rok Požar, 2026, original scientific article Abstract: A Hamilton walk in a finite graph is a walk, either open or closed, that traverses every vertex at least once. Here, we introduce Hamilton walks that are reduced in the sense that they avoid immediate backtracking: a reduced Hamilton walk never traverses the same edge forth and back consecutively. While every connected graph admits a Hamilton walk, existence of a reduced Hamilton walk is not guaranteed for all graphs. However, we prove that a reduced Hamilton walk does exist in a connected graph with minimal valency at least 2. Furthermore, given such a graph on n vertices, we present an O(n2)-time algorithm that constructs a reduced Hamilton walk of length at most n(n+3)/2. Specifically, for a graph belonging to a family of regular expander graphs, we can find a reduced Hamilton walk of length at most c(6n−2)logn+2n, where c is a constant independent of n.
Keywords: algorithm, Hamilton walk, nonstandard metric, reduced walk
Published in RUP: 09.09.2025; Views: 466; Downloads: 5
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3. Reachability relations, transitive digraphs and groupsAleksander Malnič, Primož Potočnik, Norbert Seifter, Primož Šparl, 2015, original scientific article Abstract: In [A. Malnič, D. Marušič, N. Seifter, P. Šparl and B. Zgrablič, Reachability relations in digraphs, Europ. J. Combin. 29 (2008), 1566-1581] it was shown that properties of digraphs such as growth, property ▫$\mathbf{Z}$▫, and number of ends are reflected by the properties of certain reachability relations defined on the vertices of the corresponding digraphs. In this paper we study these relations in connection with certain properties of automorphism groups of transitive digraphs. In particular, one of the main results shows that if atransitive digraph admits a nilpotent subgroup of automorphisms with finitely many orbits, then its nilpotency class and the number of orbits are closely related to particular properties of reachability relations defined on the digraphs in question. The obtained results have interesting implications for Cayley digraphs of certain types of groups such as torsion-free groups of polynomial growth.
Keywords: Cayley digraph, reachability relation
Published in RUP: 31.12.2021; Views: 2210; Downloads: 19
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7. On split liftings with sectional complementsAleksander Malnič, Rok Požar, 2018, original scientific article Keywords: algorithm, Cayley voltages, covering projection, graph, group presentation, invariant section, lifting automorphisms, linear systems over the integers, split extension
Published in RUP: 02.03.2018; Views: 4925; Downloads: 181
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9. Reachability relations in digraphsAleksander Malnič, Dragan Marušič, Norbert Seifter, Primož Šparl, Boris Zgrablić, 2008, original scientific article Abstract: In this paper we study reachability relations on vertices of digraphs, informally defined as follows. First, the weight of a walk is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed in the direction opposite to their orientation. Then, a vertex ▫$u$▫ is ▫$R_k^+$▫-related to a vertex ▫$v$▫ if there exists a 0-weighted walk from ▫$u$▫ to ▫$v$▫ whose every subwalk starting at u has weight in the interval ▫$[0,k]$▫. Similarly, a vertex ▫$u$▫ is ▫$R_k^-$▫-related to a vertex ▫$v$▫ if there exists a 0-weighted walk from ▫$u$▫ to ▫$v$▫ whose every subwalk starting at ▫$u$▫ has weight in the interval ▫$[-k,0]$▫. For all positive integers ▫$k$▫, the relations ▫$R_k^+$▫ and ▫$R_k^-$▫ are equivalence relations on the vertex set of a given digraph. We prove that, for transitive digraphs, properties of these relations are closely related to other properties such as having property ▫$\mathbb{Z}$▫, the number of ends, growth conditions, and vertex degree.
Keywords: graph theory, digraph, reachability relations, end of a graph, property ▫$\mathbb{Z}$▫, growth
Published in RUP: 03.04.2017; Views: 4225; Downloads: 147
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10. Minimal normal subgroups of transitive permutation groups of square-free degreeEdward Tauscher Dobson, Aleksander Malnič, Dragan Marušič, Lewis A. Nowitz, 2007, original scientific article Abstract: It is shown that a minimal normal subgroup of a transitive permutation group of square-free degree in its induced action is simple and quasiprimitive, with three exceptions related to ▫$A_5$▫, ▫$A_7$▫, and PSL(2,29). Moreover, it is shown that a minimal normal subgroup of a 2-closed permutation group of square-free degree in its induced action is simple. As an almost immediate consequence, it follows that a 2-closed transitive permutation group of square-free degree contains a semiregular element of prime order, thus giving a partial affirmative answer to the conjecture that all 2-closed transitive permutation groups contain such an element (see [D. Marušic, On vertex symmetric digraphs,Discrete Math. 36 (1981) 69-81; P.J. Cameron (Ed.), Problems from the fifteenth British combinatorial conference, Discrete Math. 167/168 (1997) 605-615]).
Keywords: mathematics, graph theory, transitive permutation group, 2-closed group, square-free degree, semiregular automorphism, vertex-transitive graph
Published in RUP: 03.04.2017; Views: 3952; Downloads: 100
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