All totally symmetric colored graphs

TL;DR

This paper classifies all edge-colored graphs that are fully symmetric and transitive on edges of the same color, completing previous work and confirming no such graphs exist with more than five colors.

Contribution

It provides a complete classification of all totally symmetric colored graphs, extending prior results and resolving open cases using computational methods.

Findings

No such graphs with more than 5 colors exist.

Complete classification of all totally symmetric colored graphs.

Resolved all open cases from previous research.

Abstract

In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our description completes the work done in our previous paper, where we have shown, in particular, that there are no such graphs with more than 5 colors. Using some recent results, with a help of computer, we settle all the cases that was left open in the previous paper.