On Balanced Coloring Games in Random Graphs

TL;DR

This paper investigates the thresholds of balanced coloring and Achlioptas games in random graphs, revealing differences for certain non-forest graphs and equivalences in vertex analogues.

Contribution

It demonstrates that the balanced Ramsey game and Achlioptas game have different thresholds for some non-forest graphs, answering an open question, and shows thresholds coincide in vertex analogues.

Findings

Different thresholds for balanced Ramsey and Achlioptas games on certain non-forest graphs.

Thresholds in vertex analogues are the same for all graphs F.

Settles an open question by Krivelevich et al.

Abstract

Consider the balanced Ramsey game, in which a player has r colors and where in each step r random edges of an initially empty graph on n vertices are presented. The player has to immediately assign a different color to each edge and her goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. The Achlioptas game is similar, but the player only loses when she creates a copy of F in one distinguished color. We show that there is an infinite family of non-forests F for which the balanced Ramsey game has a different threshold than the Achlioptas game, settling an open question by Krivelevich et al. We also consider the natural vertex analogues of both games and show that their thresholds coincide for all graphs F, in contrast to our results for the edge case.