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We consider an extremal problem motivated by a paper of Balogh

We consider an extremal problem motivated by a paper of Balogh [J. Balogh, A remark on the number of edge colorings of graphs, European Journal of Combinatorics 27, 2006, 565–573], who considered edge-colorings of graphs avoiding fixed subgraphs with a prescribed coloring. More precisely, given r ≥ t ≥ 2, we look for n-vertex graphs that admit the maximum number of r-edge-colorings such that at most t − 1 colors appear in edges incident with each vertex. For large n, we show that, with the exception of the case t = 2, the complete graph Kn is always the unique extremal graph. We also consider generalizations of this problem.