Paths of specified length in random k-partite graphs

Abstract

Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V i, (i=1, …,k) each having size Ω(n) and choosing each possible edge with probability p. Consider any vertex x in any V i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.