Strong games played on random graphs

TL;DR

This paper proves that in strong edge games played on large random graphs, the first player can w.h.p. secure a win in the perfect matching game, extending understanding of strategic play on probabilistic structures.

Contribution

It establishes that Red can w.h.p. win the perfect matching game on G(n,p) for large n and fixed p, providing new insights into strategic play on random graphs.

Findings

Red can w.h.p. win the perfect matching game on G(n,p)

The result holds for sufficiently large n and fixed p

Provides a probabilistic strategy for strong games 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, a perfect matching, a Hamilton cycle, etc.). It is well known that Red can always ensure at least a draw in any strong game, but finding explicit winning strategies is a difficult and a quite rare task. We consider strong games played on the edge set of a random graph G ~ G(n,p) on n vertices. We prove, for sufficiently large $n$ and a fixed constant 0 < p < 1, that Red can w.h.p win the perfect matching game on a random graph G ~ G(n,p).