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1.
Some recent discoveries about half-arc-transitive graphs : dedicated to Dragan Marušič on the occasion of his 60th birthday
Marston D. E. Conder, Primož Potočnik, Primož Šparl, 2015, original scientific article

Abstract: We present some new discoveries about graphs that are half-arc-transitive (that is, vertex- and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser ▫$D_4$▫ (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser of order 16 that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse.
Keywords: graph, edge-transitive, vertex-transitive, arc-transitive, half arc-transitive
Published in RUP: 31.12.2021; Views: 757; Downloads: 16
.pdf Full text (333,06 KB)

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On structural results of tetravalent half-arc-transitive graphs
Primož Šparl, 2018, published scientific conference contribution abstract (invited lecture)

Keywords: graph, half-arc-transitive, alternating cycle, radius
Published in RUP: 07.02.2018; Views: 2047; Downloads: 27
URL Link to full text

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Half-arc-transitive group actions with a small number of alternets
Klavdija Kutnar, 2015, published scientific conference contribution abstract

Keywords: automorphism group, half-arc-tranistive graph, alternet
Published in RUP: 15.10.2015; Views: 2313; Downloads: 28
URL Link to full text

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Recent trends and future directions in vertex-transitive graphs
Klavdija Kutnar, Dragan Marušič, 2008, original scientific article

Abstract: A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade.
Keywords: vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group
Published in RUP: 15.10.2013; Views: 3050; Downloads: 131
.pdf Full text (183,49 KB)

8.
On quartic half-arc-transitive metacirculants
Dragan Marušič, Primož Šparl, 2008, original scientific article

Abstract: Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ▫$\rho$▫ and ▫$\sigma$▫, where ▫$\rho$▫ is ▫$(m,n)$▫-semiregular for some integers ▫$m \ge 1$▫, ▫$n \ge 2▫$, and where ▫$\sigma$▫ normalizes ▫$\rho$▫, cyclically permuting the orbits of ▫$\rho$▫ in such a way that ▫$\sigma^m$▫ has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.
Keywords: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group
Published in RUP: 15.10.2013; Views: 3705; Downloads: 132
URL Link to full text

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