Strong games played on random graphs

Abstract

In a strong game played on the edge set of a graph G there are two players, Red and Blue, alternating turns in claiming previously unclaimed edges of G (with Red playing first). The winner is the first one to claim all the edges of some target structure (such as a clique K[subscript k], a perfect matching, a Hamilton cycle, etc.). In this paper we consider strong games played on the edge set of a random graph G ∼ G(n, p) on n vertices. We prove that G ∼ G(n, p) is typically such that Red can win the perfect matching game played on E(G), provided that p ∈ (0, 1) is a fixed constant.