Edge‐colorings avoiding rainbow stars

We consider an extremal problem motivated by a article 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, that is, r‐edge‐colorings avoiding rainbow‐colored stars with t edges. 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.