Graphs with many edge-colorings such that complete graphs are rainbow

Josefran de Oliveira Bastos, Carlos Hoppen, Hanno Lefmann, Andy Oertel, Dionatan Ricardo Schmidt

Abstract: We consider a version of the Erdős–Rothschild problem for families of graph patterns. For any fixed k≥3, let r0(k) be the largest integer such that the following holds for all 2≤r≤r0(k) and all sufficiently large n: The Turán graph Tk−1(n) is the unique n-vertex graph G with the maximum number of r-edge-colorings such that the edge set of any copy of Kk in G is rainbow. We use the regularity lemma of Szemerédi and linear programming to obtain a lower bound on the value of r0(k). For a more general family P of patterns of Kk, we also prove that, in order to show that the Turán graph Tk−1(n) maximizes the number of P-free r-edge-colorings over n-vertex graphs, it suffices to prove a related stability result.