Don't understand enough group theory, but two papers appear to give partial results about an open problem.

Edge colored graph isomorphism is isomorphism which preserves the edge coloring (the coloring need not be proper).

GI for circulants is polynomial. Edged-colored GI for circulants is GI complete via the simple reduction $G \to G'$.

Make a clique from $V(G)$ and color an edge $e \in E(G')$ with $1$ iff $e \in E(G)$ and $0$ otherwise. To recover $G$ from $G'$ just take the edges colored $1$.

$G \cong H \iff G' \cong H'$ where the isomorphism preserves the edge coloring.

$G'$ is edge colored clique and hence circulant.

This paper claims:

Abstract. We construct a deterministic algorithm that tests whether two circulant graphs are isomorphic. The running time is $ O(n^2 (\log n)^6 )$, where $n$ is the number of vertices of each graph. Our algorithm works for directed, undirected, and

edge-colored circulants.

The exact definition of "edge-colored" is not clear to me.

Paper proving circulant GI is polynomial in a footnote on p.1 claims:

By a graph we mean an ordinary graph, a digraph, or even an

edge colored graph

The question:

When is edge colored circulant isomorphism polynomial? What is the restriction on edge coloring?