Anti‐Ramsey numbers of doubly edge‐critical graphs

Abstract

AbstractGiven a graph H and a positive integer n, Anti‐Ramsey number AR(n, H) is the maximum number of colors in an edge‐coloring of Kn that contains no polychromatic copy of H. The anti‐Ramsey numbers were introduced in the 1970s by Erdős, Simonovits, and Sós, who among other things, determined this function for cliques. In general, few exact values of AR(n, H) are known. Let us call a graph H doubly edge‐critical if χ(H−e)≥p+ 1 for each edge e∈E(H) and there exist two edges e1, e2 of H for which χ(H−e1−e2)=p. Here, we obtain the exact value of AR(n, H) for any doubly edge‐critical H when n⩾n0(H) is sufficiently large. A main ingredient of our proof is the stability theorem of Erdős and Simonovits for the Turán problem. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 210–218, 2009