Tetravalent edge-transitive graphs of girth at most 4

Abstract

This is the first in the series of articles stemming from a systematical investigation of finite edge-transitive tetravalent graphs, undertaken recently by the authors. In this article, we study a special but important case in which the girth of such graphs is at most 4. In particular, we show that, except for a single arc-transitive graph on 14 vertices, every edge-transitive tetravalent graph of girth at most 4 is the skeleton of an arctransitive map on the torus or has one of these two properties:(1) there exist two vertices sharing the same neighbourhood, (2) every edge belongs to exactly one girth cycle.Graphs with property (1) or (2) are then studied further. It is shown that they all arise either as subdivided doubles of smaller arc-transitive tetravalent graphs, or as line graphs of triangle-free (G, 1)-regular and (G, 2)-arc-transitive cubic graphs, or as partial line graphs of certain cycle decompositions of smaller tetravalent graphs.

A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

“…The results of this paper confirm that the list given in [31] contains all 4-GHATs of order at most 100.…”

“…We finish this section by mentioning an earlier attempt of Stephen Wilson and the first author of this paper to compile a census of all small edge-transitive graphs of valence 4, and thus, in particular, of all 4-GHATs; see [31]. The results of this paper confirm that the list given in [31] contains all 4-GHATs of order at most 100.…”

Linking rings structures and semisymmetric graphs: Cayley constructions

“…This paper continues work from [13,14], where the present authors introduced linking rings structures, that is, certain decompositions of tetravalent vertex-transitive graphs into cycles (see Definition 1.1 for a precise definition). Our original motivation for studying such structures stems from the desire to understand tetravalent semisymmetric graphs.…”

Locally arc-transitive graphs of valence {3,4} with trivial edge kernel

“…This will be proved by means of the socalled subdivided doubles, a construction that was introduced in [22] and that can be described as follows.…”