Waiter-Client Triangle-Factor Game on the Edges of the Complete Graph

TL;DR

This paper confirms a conjecture about the exact length of a Waiter-Client game on the edges of a complete graph, where Waiter aims to quickly force a triangle-factor in Client's graph.

Contribution

The paper proves the conjectured asymptotic value of the game length, settling a previously open question about optimal strategies in this combinatorial game.

Findings

Confirmed the conjecture that the game lasts approximately 7/6 times n rounds.

Established the asymptotic bounds for the game length.

Validated the optimality of strategies for both players.

Abstract

Consider the following game played by two players, called Waiter and Client, on the edges of $K_n$ (where $n$ is divisible by $3$). Initially, all the edges are unclaimed. In each round, Waiter picks two yet unclaimed edges. Client then chooses one of these two edges to be added to Waiter's graph and one to be added to Client's graph. Waiter wins if she forces Client to create a $K_3$-factor in Client's graph at some point, while if she does not manage to do that, Client wins. It is not difficult to see that for large enough $n$, Waiter has a winning strategy. The question considered by Clemens et al. is how long the game will last if Waiter aims to win as soon as possible, Client aims to delay her as much as possible, and both players play optimally. Denote this optimal number of rounds by $\tau_{WC}(\mathcal{F}_{n,K_3-\text{fac}},1 ) $. Clemens et al. proved that $\frac{13}{12}n \leq \tau_{WC}(\mathcal{F}_{n,K_3-\text{fac}},1 ) \leq \frac{7}{6}n+o(n) $, and conjectured that $\tau_{WC}(\mathcal{F}_{n,K_3-\text{fac}},1 ) = \frac{7}{6}n+o(n) $. In this note, we verify their conjecture.