Tetravalent edge-transitive graphs of girth at most 4

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 arc-transitive map on the torus or has one of these two properties:(1) exist two vertices sharing the same neighbourhood, edge belongs to exactly one girth cycle. s 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.