Recipes for Edge-Transitive Tetravalent Graphs

This paper is to accompany the Census of Edge-Transitive Tetravalent Graphs, available at jan.ucc.nau.edu/~swilson/C4FullSite/index.html, which is a collection of all known edge-transitive graphs of valence 4 up to 512 vertices. The Census contains information for each graph. This information includes parameters such as group order, diameter, girth etc., all known constructions, relations to other graphs in the Census and intersting substructures such as colorings, cycle structures, and dissections. We try to present most graphs as members of one or more parameterized families, and one purpose of this paper is to gather together, here in one place, descriptions of each of these families, to show how each is constructed, what the history of each is and how one family is related to another. We also discuss in this paper the theory and techniques behind computer searches leading to many entries in the Census.