Coloring the Edges of a Random Graph without a Monochromatic Giant Component

Abstract

We study the following two problems: i) Given a random graph $G_{n, m}$ (a graph drawn uniformly at random from all graphs on $n$ vertices with exactly $m$ edges), can we color its edges with $r$ colors such that no color class contains a component of size $\Theta(n)$? ii) Given a random graph $G_{n,m}$ with a random partition of its edge set into sets of size $r$, can we color its edges with $r$ colors subject to the restriction that every color is used for exactly one edge in every set of the partition such that no color class contains a component of size $\Theta(n)$?

We prove that for any fixed $r\geq 2$, in both settings the (sharp) threshold for the existence of such a coloring coincides with the known threshold for $r$-orientability of $G_{n,m}$, which is at $m=rc_r^*n$ for some analytically computable constant $c_r^*$. The fact that the two problems have the same threshold is in contrast with known results for the two corresponding Achlioptas-type problems.