On classifying finite edge colored graphs with two transitive automorphism groups

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with 1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes. The classification of finite simple groups in the 1980s has led to classification theorems concerning a variety of designs and geometric structures. Edge colored graphs generalize balanced incomplete block designs with 1 and one-factorizations of complete graphs and provide a reinterpretation of metric spaces. This paper classifies the doubly transitive edge colored graphs (abbreviated 2-t ec-graphs), extending results of Kantor [14] and Cameron and Korchmaros [8]. The doubly transitive symmetric graph designs of Cameron [7] when 1 match the 2-t ec-graphs for which the number of colors equals the number of vertices. Edge colored graphs, which in this article are always colorings of complete graphs, are closely related to the rainbows Aschbacher defined in