Totally symmetric colored graphs

Abstract

Abstract:In this paper we describe almost all edge-colored complete graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. This generalizes the recent description of self-complementary symmetric graphs by Peisert and gives examples of permutation groups that require more than 5 colors to be represented as the automorphism group of a k-colored graph. This also contributes to the recent study of homogeneous factorizations of complete graphs. The result relies on the classification of finite simple groups.

Decompositions of the automorphism groups of edge-colored graphs into the direct product of permutation groups

“…Various aspects of this general problem are investigated. For example, it leads to the concept of colored totally symmetric graphs, that was described in [11,12]. This coincides to a large extent with the research on homogeneous factorization of graphs (c.f., [4,15,16]).…”

The graphical complexity of direct products of permutation groups

“…In [9], it was shown that if two permutation groups (A, V) and (B, W) belong to GR(k), then the direct sum (A⊕B, V ∪W) belongs to GR(k +1), and all cases where it belongs to GR(k) have been described. This led to a description of the so-called totally symmetric permutation groups, which appeared in [10]. For each k ≤ 6 an example of a permutation group with graphical complexity k was found.…”

Graphical cyclic permutation groups

“…Similar results are obtained for other products. In [10], we have described almost all k-colored graphs with the highest degree of symmetry. In particular, we have shown that if a k-colored graph is edge-transitive and colorsymmetric, then k ≤ 5.…”