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

## TL;DR

This paper investigates conditions under which the direct product of two permutation groups, where at least one is not an automorphism group of an edge-colored graph, results in an automorphism group of an edge-colored graph or digraph.

## Contribution

It provides necessary and sufficient conditions for the direct product of permutation groups to be an automorphism group of an edge-colored graph or digraph, extending previous results.

## Key findings

- Identifies conditions for the product to be an automorphism group of an edge-colored graph.
- Extends the analysis to edge-colored digraphs.
- Provides a complete characterization for such group products.

## Abstract

In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jez, A. Kisielewicz have proved that the direct product of automorphism groups of edge-colored graphs is itself the automorphism groups of an edge-colored graph. In this paper, we study the direct product of two permutation groups such that at least one of them fails to be the automorphism group of an edge-colored graph. We find necessary and sufficient conditions for the direct product to be the automorphism group of an edge-colored graph. The same problem is solved for the edge-colored digraphs.