A remark on the number of edge colorings of graphs

• Authors: József Balogh
• Venue: European Journal of Combinatorics
• Year: 2006

Fix a 2-coloring Hk + 1 of the edges of a complete graph Kk + 1. Let C(n, Hk + 1) denote the maximum possible number of distinct edge-colorings of a simple graph on n vertices with two colors, which contain no copy of Kk + 1 colored exactly as Hk + 1. It is shown that for every fixed k and all n n0(k), if in the colored graph Hk + 1 both colors were used, then C(n, Hk + 1) = 2tk(n), where tk(n) is the maximum possible number of edges of a graph on n vertices containing no K k + 1. The proofs are based on Szemerédi's Regularity Lemma together with the Simonovits Stability Theorem, and provide one of the growing number of examples of a precise result proved by applying the Regularity Lemma.