Rainbow connectivity Maker-Breaker game

TL;DR

This paper analyzes biased Maker-Breaker games on graphs, focusing on rainbow connectivity and spanning trees, determining threshold biases, and exploring strategies that combine randomized and balancing approaches.

Contribution

It provides a detailed analysis of the rainbow connectivity Maker-Breaker game, determines threshold biases on complete graphs, and disproves a previous conjecture.

Findings

Threshold bias for rainbow connectivity on complete graphs is established.

A Maker's strategy combining randomized and balancing techniques is developed.

The order of the threshold bias for the diameter game is determined.

Abstract

We study biased Maker-Breaker games on a graph system $\{G_1,\ldots,G_s\}$, in which Maker's goal is to claim certain rainbow structures, i.e., specified subgraphs consisting of at most one edge from each graph $G_i$. We consider the rainbow-connectivity game, in which Maker wants to claim a rainbow path between every pair of vertices. We analyse this game in detail, essentially determining the threshold bias when played on the system of complete graphs, and observing that whether the random graph intuition holds depends on the size of $s$. The key ingredient of our result is the analysis of a Maker's strategy that combines several randomized strategies with an appropriately designed balancing game. As a byproduct, we find the order of the threshold bias for the Maker-Breaker diameter game, and disprove a conjecture by Balogh, Martin and Pluh\'ar. Another natural and general way to analyse Maker-Breaker games that are played on a colored board is to require Maker to occupy a rainbow winning set of a given positional game. In the case of the connectivity game, Maker's goal is to claim a rainbow spanning tree. For this game played on the system of complete graphs, we establish matching upper and lower bounds on the threshold bias, up to constant factors.