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  • Almost all quartic half-arc-transitive weak metacirculants of Class II are of Class IV
    Šparl, Primož
    A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. A weak metacirculant is a graph admitting a transitive metacyclic group that is a group generated by two automorphisms ... ▫$\rho$▫ and ▫$\sigma$▫, where ▫$\rho$▫ is ▫$(m,n)$▫-semiregular for some integers ▫$m \ge 1$▫ and ▫$n \ge 2$▫, and where ▫$\sigma$▫ normalizes ▫$\rho$▫. It was shown in [D. Marušič, P. Šparl, On quartic half-arc-transitive metacirculants, J. Algebr. Comb. 28 (2008) 365-395] that each connected quartic half-arc-transitive weak metacirculant ▫$X$▫ belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph ▫$X_\rho$▫ relative to the semiregular automorphism ▫$\rho$▫. The first of these classes, called Class I, coincides with the class of so-called tightly attached graphs. Class II consists of the quartic half-arc-transitive weak metacirculants for which the quotient graph ▫$X_\rho$▫ is a cycle with a loop at each vertex. Class III consists of those graphs for which each vertex of the quotient graph ▫$X_\rho$▫ is connected to three other vertices, to one with a double edge. Finally, Class IV consists of those graphs for which ▫$X_\rho$▫ is a simple quartic graph. This paper consists of two results concerning graphs of Class II. It is shown that, with the exception of the Doyle-Holt graph and its canonical double cover, each quartic half-arc-transitive weak metacirculant of Class II is also of Class IV. It is also shown that although quartic half-arc-transitive weak metacirculants of Class II which are not tightly attached exist they are "almost tightly attached". More precisely, their radius is at most four times their attachment number.
    Vir: Algebraic and topological graph theory (Str. 1737-1742)
    Vrsta gradiva - prispevek na konferenci
    Leto - 2010
    Jezik - angleški
    COBISS.SI-ID - 15369561

    Povezava(-e):

    http://dx.doi.org/10.1016/j.disc.2009.11.015
    DOI

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