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 $K_k$, a perfect matching, a Hamilton cycle, etc.).
In this paper we consider strong games played on the edge set of a random graph $G\sim G(n,p)$ on $n$ vertices. We prove that $G\sim G(n,p)$ is typically such that Red can win the perfect matching game played on $E(G)$, provided that $p\in(0,1)$ is a fixed constant.