An upper bound on the extremal version of Hajnal's triangle-free game

TL;DR

This paper establishes an upper bound on the maximum number of edges in a triangle-free graph formed during a game where two players alternately add edges without creating triangles, with the aim of either ending or prolonging the game.

Contribution

The paper provides a new upper bound on the extremal value of the triangle-free game saturation number under optimal play.

Findings

Proved an upper bound for the $K_3$ game saturation number.

Analyzed the strategic constraints of triangle-free graph construction.

Contributed to understanding extremal limits in combinatorial game theory.

Abstract

A game starts with the empty graph on $n$ vertices, and two player alternate adding edges to the graph. Only moves which do not create a triangle are valid. The game ends when a maximal triangle-free graph is reached. The goal of one player is to end the game as soon as possible, while the other player is trying to prolong the game. With optimal play, the length of the game (number of edges played) is called the $K_3$ game saturation number. In this paper we prove an upper bound for this number.