18 research outputs found

    On tetravalent half-arc-transitive graphs of girth 5

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    A subgroup of the automorphism group of a graph \G is said to be {\em half-arc-transitive} on \G if its action on \G is transitive on the vertex set of \G and on the edge set of \G but not on the arc set of \G. Tetravalent graphs of girths 33 and 44 admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth 55. We show that, with two exceptions, all such graphs only have directed 55-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed 55-cycles, study some of their graph theoretic properties and prove that the 55-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth 55 admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth 55

    Resolution of a conjecture about linking ring structures

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    An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Poto\v{c}nik and S. Wilson, titled `Linking rings structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7 (2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper

    Recent trends and future directions in vertex-transitive graphs

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    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
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