Coloring the edges of a random graph without a monochromatic giant component

Abstract

Abstract Our goal is to color the edges of a random graph G n , m (a graph drawn uniformly at random from all graphs on n vertices with exactly m edges) with a fixed number r of colors such that no color class induces a component of size Ω ( n ) – a so called ‘giant component’. We prove that for every r ⩾ 2 there exists an analytically computable constant c r ∗ for which the following holds: For any c c r ∗ , with probability 1 − o ( 1 ) there exists an r-edge-coloring of G n , rcn in which every monochromatic component has sublinearly many vertices. On the other hand, for any c > c r ∗ , with probability 1 − o ( 1 ) every r-edge-coloring of G n , rcn contains a monochromatic component on linearly many vertices. In other words, we prove that the property in question has a sharp threshold at m = r c r ∗ n .