Chromatic number, clique subdivisions, and the conjectures of Hajos and Erdos-Fajtlowicz

Abstract

For a graph G, let (G) denote its chromatic number and (G) denote the order of the largest clique subdivision in G. Let H(n) be the maximum of (G)= (G) over all n-vertex graphs G. A famous conjecture of Haj os from 1961 states that (G) (G) for every graph G. That is, H(n) 1 for all positive integers n. This conjecture was disproved by Catlin in 1979. Erd}os and Fajtlowicz further showed by considering a random graph that H(n) cn1=2= log n for some absolute constant c > 0. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant C such that (G)= (G) Cn1=2= log n for all n-vertex graphs G. In this paper we prove the Erd}os-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can nd in every graph on n vertices with independence number .National Science Foundation (U.S.) (NSF grant DMS-110118)National Science Foundation (U.S.) (CAREER award DMS-0812005)United States-Israel Binational Science Foundatio