The $K^4$-Game

TL;DR

This paper analyzes the $K^4$-building game on an infinite complete graph, providing a winning strategy for the first player to claim all edges of a 4-vertex subset, thus guaranteeing victory.

Contribution

The paper introduces a novel winning strategy for the first player in the $K^4$-game on an infinite graph, establishing a guaranteed win.

Findings

First player can always win with the proposed strategy.

The strategy guarantees victory regardless of the opponent's moves.

The game dynamics on an infinite graph are fully characterized.

Abstract

We investigate a two player game called the $K^4$-building game: two players alternately claim edges of an infinite complete graph. Each player's aim is to claim all six edges on some vertex set of size four for themself. The first player to accomplish this goal is declared the winner of the game. We present a winning strategy which guarantees a win for the first player.