Waiter–Client triangle-factor game on the edges of the complete graph

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 $\frac{13}{12}n\le {\tau }_{WC}({\mathcal{F}}_{n,K_3-\text{fac}},1)\le \frac{7}{6}n+o\left(n\right)$, and conjectured that ${\tau }_{WC}({\mathcal{F}}_{n,K_3-\text{fac}},1)=\frac{7}{6}n+o\left(n\right)$. In this note, we verify their conjecture.

Introduction

Positional games are a large class of two-player combinatorial games characterized by the following setting. We have a (usually finite) set $X$, called a board; a family of subsets of $X$, called winning sets; and a rule determining which player wins the game. These games attracted wide attention, starting with the papers of Hales and Jewett [9] and Erdős and Selfridge [7]. We refer the reader interested in positional games to the book of Hefetz, Krivelevich, Stojaković and Szabó [11].

Probably the most studied type of positional games are the so-called Maker–Breaker games. These games are played by two players, called Maker and Breaker, as follows. We are given an integer $b\ge 1$, a set $X$ and a family of winning sets $\mathcal{F}$ of the subsets of $X$. The players alternately claim previously unclaimed elements of $X$, with Maker claiming one element in each round and Breaker claiming $b$ elements in each round. Maker wins if she manages to claim any $F\in \mathcal{F}$, while Breaker wins if he prevents Maker from doing so. We are usually interested in the two main questions: which player has a winning strategy for given $X,b,\mathcal{F}$? And if Maker has a winning strategy, how fast can she win? Many variants of Maker–Breaker games were considered, see for instance [3], [5], [6], [10], [15], [16].

Another rather similar type of the well studied positional games are the Waiter–Client games, introduced originally in the book of Beck [1] under the name Picker–Chooser games. These games are played by two players, called Waiter and Client, in the following manner. As in the Maker–Breaker games, we are given an integer $b\ge 1$, a set $X$ and a family of winning sets $\mathcal{F}$ of the subsets of $X$. In each round, Waiter picks $b+1$ previously unclaimed elements of $X$ and offers them to Client. Client chooses one of these elements and adds it to his graph, while the remaining $b$ elements become a part of Waiter’s graph. Waiter wins if she forces Client to create a winning set $F\in \mathcal{F}$ in Client’s graph. If Client can prevent that, he wins. Once again, the questions that interest us are: for given $X,b,\mathcal{F}$, which player has a winning strategy? And if Waiter can win, how fast can she guarantee her victory to be? These problems are well studied in many cases, see for instance [2], [12], [13], [14], [17].

Assume that for our triple $X,b,\mathcal{F}$, Waiter wins the corresponding Waiter–Client game. Then we will denote by ${\tau }_{WC}(\mathcal{F},b)$ the number of rounds of the game when Waiter tries to win as fast as possible, Client tries to slow her down as much as possible, and they both play optimally. What the ground set $X$ is will usually be clear from the context.

When $b=1$, we call the corresponding Waiter–Client game unbiased. Recently, Clemens et al. [4] studied several unbiased Waiter–Client games played on the edges of the complete graph, i.e. with $X=E\left(K_n\right)$. For $n$ divisible by $3$, they considered the triangle-factor game, where the winning sets are the collections of $\frac{n}{3}$ vertex disjoint triangles. It is not hard to verify that for $n$ large enough, Waiter can win this game. Moreover, Clemens et al. obtained the following theorem giving the lower and upper bounds on the optimal duration of the game.

Theorem 1.1

Assume $n$ is divisible by $3$ and large enough that Waiter wins the corresponding unbiased triangle-factor game on the edges of $K_n$. Then

$$\frac{13}{12}n\le {\tau }_{WC}({\mathcal{F}}_{n,K_3-\text{fac}},1)\le \frac{7}{6}n+o\left(n\right).$$

Further, they made a conjecture that ${\tau }_{WC}({\mathcal{F}}_{n,K_3-\text{fac}},1)=\frac{7}{6}n+o\left(n\right)$. Our aim in this note is to improve the lower bound from $\frac{13}{12}n$ to $\frac{7}{6}n$ and hence to verify their conjecture.

Theorem 1.2

Assume $n$ is divisible by $3$ and large enough that Waiter wins the corresponding unbiased triangle-factor game on the edges of $K_n$. Then

$${\tau }_{WC}({\mathcal{F}}_{n,K_3-\text{fac}},1)\ge \frac{7}{6}n.$$

Finally, let us note that unbiased triangle-factor game on the edges of $K_n$ is an example of a more general phenomena that for a given board and parameters, Waiter can typically win the corresponding Waiter–Client game asymptotically at least as fast as Maker can win the corresponding Maker–Breaker game. Indeed, we know that in this case, Waiter needs $\frac{7}{6}n+o\left(n\right)$ rounds if she plays optimally, while it was observed by Krivelevich and Szabó that Maker cannot win the Maker–Breaker version of the game in less than $\frac{7}{6}n$ rounds (see [8]). The optimal number of rounds for the Maker–Breaker variant is still not known. For a more thorough discussion of this relation between Maker–Breaker and Waiter–Client games, we refer reader to the paper of Clemens et al. [4].

The rest of the paper is organized as follows: in Section 2, we very briefly summarize some notation that we will use. Then we set up the necessary definitions and prove some easy results about these in Section 3. After that, we prove Theorem 1.2 in Section 4, by giving a strategy of Client and analyzing the game when Client uses this strategy.