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Opis: The Gray configuration is a (27_3) configuration which typically is realized as the points and lines of the 3×3×3 integer lattice. It occurs as a member of an infinite family of configurations defined by Bouwer in 1972. Since their discovery, both the Gray configuration and its Levi graph (i.e., its point-line incidence graph) have been the subject of intensive study. Its automorphism group contains cyclic subgroups isomorphic to Z3 and Z9, so it is natural to ask whether the Gray configuration can be realized in the plane with any of the corresponding rotational symmetry. In this paper, we show that there are two distinct polycyclic realizations with Z3 symmetry. In contrast, the only geometric polycyclic realization with straight lines and Z9 symmetry is only a “weak” realization, with extra unwanted incidences (in particular, the realization is actually a (27_4) configuration).
Ključne besede: Gray graph, Gray configuration, polycirculant, polycyclic configuration
Objavljeno v RUP: 29.09.2025; Ogledov: 413; Prenosov: 4
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Opis: A graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length. While the tetravalent edge-transitive graphs admitting a semiregular automorphism with only one orbit are easy to determine, those that admit a semiregular automorphism with two orbits took a considerable effort and were finally classified in 2012. Of the several possible different "types" of potential tetravalent edge-transitive graphs admitting a semiregular automorphism with three orbits, only one "type" has thus far received no attention. In this paper we focus on this class of graphs, which we call the Woolly Hat graphs. We prove that there are in fact no edge-transitive Woolly Hat graphs and classify the vertex-transitive ones.
Ključne besede: edge-transitive, vertex-transitive, tricirculant, Woolly Hat graphs
Objavljeno v RUP: 10.09.2025; Ogledov: 515; Prenosov: 7
Celotno besedilo (552,80 KB)

Ključne besede: map, oriented map, truncation, medial, snub, flag graph, arc graph
Objavljeno v RUP: 16.03.2018; Ogledov: 4401; Prenosov: 174
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