On Gallai $d$-colored circulants

Abstract

A $d$-graph is a complete graph whose edges are colored with colors from $[d]$. It is Gallai if it contains no triangle whose three edges have distinct colors. We prove that every Gallai $d$-circulant is an iterated lexicographic product of circulants using at most two colors. The proof uses the compatibility of the known canonical modular decomposition with the cyclic action. For $d$-graphs avoiding both a rainbow triangle $Δ$ and the two-colored four-vertex graph $Π$ whose chromatic components are paths, we give a more explicit classification. Color-preserving vertex transitivity is equivalent to circular symmetry and to a reduced positional game tree whose branching numbers and labels depend only on depth. The edge colors in the given cyclic labeling are determined by a chain of divisors of the order. This description yields a complete isomorphism invariant and a linear-time recognition procedure in the difference-word representation. The characterization of CIS $d$-circulants reduces to CIS two-colored factors. Finally, we extend the numerical CIS criterion to $d$-graphs whose chromatic components are vertex-transitive.