Edge‐colorings avoiding rainbow stars

Abstract

Abstract 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 , we look for n ‐vertex graphs that admit the maximum number of r ‐edge‐colorings such that at most 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 , the complete graph is always the unique extremal graph. We also consider generalizations of this problem.