The diameter game

TL;DR

This paper introduces the $d$-diameter game on complete graphs, analyzing the winning strategies of Maker and Breaker under various biases, revealing surprising thresholds for winning conditions based on the number of edges each player can choose per turn.

Contribution

It presents the first analysis of the $d$-diameter game, especially the surprising result that Maker's winning threshold changes with the number of edges he can select per turn.

Findings

Breaker wins the 2-diameter game with one edge per turn.

Maker wins the 2-diameter game if allowed two edges per turn.

Threshold bias for Breaker to win in the biased game is approximately $(1/9)n^{1/8}/( ext{ln} n)^{3/8}$.

Abstract

A large class of Positional Games are defined on the complete graph on $n$ vertices. The players, Maker and Breaker, take the edges of the graph in turns, and Maker wins iff his subgraph has a given -- usually monotone -- property. Here we introduce the $d$-diameter game, which means that Maker wins iff the diameter of his subgraph is at most $d$. We investigate the biased version of the game; i.e., when the players may take more than one, and not necessarily the same number of edges, in a turn. Our main result is that we proved that the $2$-diameter game has the following surprising property: Breaker wins the game in which each player chooses one edge per turn, but Maker wins as long as he is permitted to choose $2$ edges in each turn whereas Breaker can choose as many as $(1/9)n^{1/8}/(\ln n)^{3/8}$. In addition, we investigate $d$-diameter games for $d\ge 3$. The diameter games are strongly related to the degree games. Thus, we also provide a generalization of the fair degree game for the biased case.