Almost all quartic half-arc-transitive weak metacirculants of Class II are of Class IV

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 ρρ and σσ, where ρρ is (m,n)(m,n)-semiregular for some integers m≥1m≥1 and n≥2n≥2, and where σσ normalizes ρρ. 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 XX belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph XρXρ relative to the semiregular automorphism ρρ. 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 XρXρ is a cycle with a loop at each vertex. Class III consists of those graphs for which each vertex of the quotient graph XρXρ is connected to three other vertices, to one with a double edge. Finally, Class IV consists of those graphs for which XρXρ 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.