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 K n is always the unique extremal graph. We also consider generalizations of this problem.