Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Creating spanning trees in Waiter-Client games

Grzegorz Adamski, Sylwia Antoniuk, Małgorzata Bednarska-Bzdęga, Dennis Clemens, Fabian Hamann, Yannick Mogge

TL;DR

It is proved that for every constant $c<1/3$, if $\Delta(T_n)\le cn$ and $n$ is sufficiently large, then Waiter has a winning strategy in $\textrm{WC}(n,T_n)$.

Abstract

For a positive integer $n$ and a tree $T_n$ on $n$ vertices, we consider an unbiased Waiter-Client game $\textrm{WC}(n,T_n)$ played on the complete graph~$K_n$, in which Waiter's goal is to force Client to build a copy of $T_n$. We prove that for every constant $c<1/3$, if $Δ(T_n)\le cn$ and $n$ is sufficiently large, then Waiter has a winning strategy in $\textrm{WC}(n,T_n)$. On the other hand, we show that there exist a positive constant $c'<1/2$ and a family of trees $T_{n}$ with $Δ(T_n)\le c'n$ such that Client has a winning strategy in the $\textrm{WC}(n,T_n)$ game for every $n$ sufficiently large. We also consider the corresponding problem in the Client-Waiter version of the game.

Creating spanning trees in Waiter-Client games

TL;DR

It is proved that for every constant , if and is sufficiently large, then Waiter has a winning strategy in .

Abstract

For a positive integer and a tree on vertices, we consider an unbiased Waiter-Client game played on the complete graph~, in which Waiter's goal is to force Client to build a copy of . We prove that for every constant , if and is sufficiently large, then Waiter has a winning strategy in . On the other hand, we show that there exist a positive constant and a family of trees with such that Client has a winning strategy in the game for every sufficiently large. We also consider the corresponding problem in the Client-Waiter version of the game.
Paper Structure (4 sections, 3 theorems)

This paper contains 4 sections, 3 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Probabilistic tools In our probabilistic argument, we will use a variant of the Chernoff bound, for a sum of independent random variables $Z_i$ such that the number of terms is a random variable, not necessarily independent of $Z_i$. For $n\in{\mathbb N}$ and $\rho\in(0,1)$, let $(Z_i)_{i=1}^{n}$ be a sequence of i.i.d. random variables with probability distribution Z_i = 1,\text{with probability } \rho,0,\text{with probability } 1-\rho. Then for any random variable $T$ taking values in the set $\{0,1,\dots,n\}$, we have \mathbb{P}\left(\left|\sum_{i=1}^T Z_i - T\rho \right| > 2\sqrt{n\log n} \right) = o(n^{-2}). Using the union bound and the Chernoff inequality (see e.g. janson2011random), we obtain \mathbb{P}\left(\left|\sum_{i=1}^T Z_i - T\rho \right| > 2\sqrt{n\log n} \right)\leq \sum_{k=1 }^n \mathbb{P}\left(\left|\sum_{i=1}^k Z_i - k\rho \right| > 2\sqrt{n\log n} \right) \leq\leq \sum_{k=1}^n 2\exp\left(\frac{-8n\log n}{k}\right) \leq 2n\exp(-8\log n) = o(n^{-2}). We say that an event, depending on $n$, holds asymptotically almost surely (a.a.s.) if it holds with probability tending to 1 if $n$ tends to infinity. We will use a variant of the Erdős-Selfridge Breaker's winning criterion (see e.g. beck2008combinatorialhefetz2014positional), adapted to the Client-Waiter version by the third author, which can be also applied for Waiter-Client games. Consider a Waiter-Client game on a hypergraph $(\mathcal{X},\mathcal{F})$ satisfying $$∑_F∈ F 2^-|F| + 1 < 1.$$ Then Waiter has a strategy to force Client to claim at least one element in each of the sets in $\mathcal{F}$. For the proof of Theorem \ref{['thm:giventree']} we aim to provide Waiter's strategy that forces Client to build any fixed $n$-vertex tree $T$ whose maximum degree is at most $\left(\frac{1}{3}-\varepsilon\right)n$. For this, we will distinguish trees by certain structural properties, but in any case, the overall strategy for Waiter will then be to first force a red copy of a subforest $F\subseteq T$ which contains all vertices of large degree in $T$, and then she will complete this subforest to a copy of $T$ while making use of the structural properties mentioned above. Note that in such a procedure, when $F$ is already embedded, we also fix the images of some of the vertices in order to be able to embed the remaining forest $T\setminus E(F)$. This is why we need to study games in which Waiter wants to force rooted forests. Below we prepare the tools for our main strategy, which is given in Section \ref{['sec:proof.linear.degree']}, by collecting and proving several lemmas regarding forcing different kinds of forests and forests with roots. A rich collection of lemmas is used to prove Theorems \ref{['thm:root1del']} and \ref{['thm:doubleroot3']} -- only these two results, together with Lemma \ref{['lem:doubleroot1v']}, are applied in the proof of Theorem \ref{['thm:giventree']}. The following two lemmas are more quantitative versions of results from clemens2020fast on forcing Hamilton paths with fixed endpoints. For $n\geq 5$, Waiter can force Client to build a red Hamilton path on $K_n$ within $n-1$ rounds and such that one of its endpoints is incident with at most two blue edges. Let $v$ be any vertex of $K_n$. In each of the first two rounds Waiter offers any two free edges incident with $v$. The result is a red path $P$ on three vertices whose endpoints are not incident with any other colored edge outside the path. Next, in each of the following $n-3$ rounds, let $P$ be the current red path, and let $w$ be any vertex not in $P$. Then Waiter offers the two edges between $w$ and the endpoints of $P$, hence extending the red path $P$ by one vertex. By the end of round $n-1$, the path $P$ is a Hamilton path. Let $w$ be the vertex which was added in the last round. Then there are at most two blue edges incident with $w$: the blue edge that was offered in the last round, and maybe one more edge if $vw$ was offered within the first two rounds. Let $x,y\in V(K_n)$ with $n\geq 8$. Then within $n$ rounds Waiter can force Client to build a red Hamilton path between $x$ and $y$ in $K_n$. Using Lemma \ref{['lem:Ham.path.simple']}, within $n-3$ first rounds Waiter can force a red Hamilton path $P=(v_1,v_2,\ldots,v_{n-2})$ on $K_n-\{x,y\}$ in which $v_1$ is incident with at most two blue edges. In round $n-2$ she offers the edges $xv_{n-2}$ and $yv_{n-2}$, and without loss of generality we can assume that Client claims $v_{n-2}y$. Let $i,j$, with $3\leq i,j\leq n-2$, be such that $v_1v_i$ and $v_1v_j$ are free. This is possible since so far $v_1$ is incident with at most two blue edges. Then in round $n-1$ Waiter offers edges $v_1v_i$ and $v_1v_j$, and again without loss of generality we can assume that Client claims the edge $v_1v_i$. Finally, in round $n$, Waiter offers the edges $xv_1$ and $xv_{i-1}$. It is easy to see that no matter which edge Client claims, a red Hamilton path is built as required. The next lemma shows that Waiter can force quickly a perfect matching in an almost complete bipartite graph (with equal bipartition). Let $t\ge 4$ and $K_{t,t}^{-}$ be the graph obtained from the complete bipartite graph $K_{t,t}$ by removing one edge. By offering edges of $K_{t,t}^-$ only, Waiter can force a red copy of a perfect matching in $K_{t,t}^{-}$ within $t+1$ rounds. In order to build a perfect matching in $K_{4,4}^{-}$, in each round Waiter offers the two dashed edges. We will use induction. First consider the case $t=4$. Let $\{u_1,u_2,u_3,u_4\}$ and $\{v_1,v_2,v_3,v_4\}$ be the bipartition classes of $K_{4,4}^-$. Assume that $u_1v_1$ is the missing edge. Waiter plays in such a way that in each round Client's choices are irrelevant for the obtained structure of red and blue graphs, but they do have impact on the vertex labels in the edges offered by Waiter. (see Figure \ref{['Fig:M_4vs.K_4,4']}). Waiter starts by offering the edges $u_1v_2$ and $u_2v_1$. By symmetry we can assume that $u_1v_2$ becomes red. Then Waiter offers the edges $u_1v_3$ and $u_1v_4$. Client chooses one of them, say $u_1v_3$. Then Waiter offers the edges $u_3v_1$ and $u_4v_1$. Again we can assume that Client colors for instance $u_3v_1$ red. Next Waiter offers the edges $u_2v_4$ and $u_4v_4$. Without loss of generality Client colors $u_2v_4$ red. Finally, Waiter offers the edges $u_4v_2$ and $u_4v_3$. Client colors any of them red, say $u_4v_2$, and we get the desired red matching, namely $\{u_1v_3, u_2v_4, u_3v_1, u_4v_2\}$, within 5 rounds. Now for $t>4$, suppose that $uv$ is the missing edge. Then in the first round Waiter offers one edge incident with $u$, and the other incident with $v$, say $uv'$ and $vu'$. By symmetry we can assume that Client colors $uv'$ red. Then we remove from our graph the vertices $u$ and $v'$ and the blue edge $vu'$, and the problem reduces to forcing a perfect matching in $K_{t-1,t-1}^-$ (now $vu'$ is the missing edge), which by inductive assumption can be done within $t$ rounds. Altogether, we get a perfect matching in $K_{t,t}^-$ within $t+1$ rounds. We begin by defining a couple of technical notions used extensively throughout the remaining part of this section. Let $F_1,F_2,\ldots,F_{\ell}$ be vertex-disjoint rooted forests (i.e. each component has exactly one root), such that $e(F_i)>0$ for some $i$. We call the tuple $F=(F_1,F_2,\ldots,F_{\ell})$ a rooted forest-tuple, and define $V(F) = \bigcup_{i\in[\ell]} V(F_i)$, $v(F) = |V(F)|$, $E(F) = \bigcup_{i\in[\ell]} E(F_i)$, $e(F) = |E(F)|$, and a(F)= \max \{ e(F_i):~ i\in [\ell]\},k(F)= |\{i\in [\ell]:~ e(F_i)=a(F)\}|,t(F)= |\{i\in [\ell]: e(F_i)>0\}|. Moreover, we define the following: Let $\ell'\le \ell$. We call a rooted forest-tuple $F'=(F_1',F_2',\ldots,F_{\ell'}')$ a valid subforest of $F$ if for every $i\in [\ell']$, we have $F_i'\subseteq F_i$ and every root of $F_i'$ is a root of $F_i$. Furthermore, we assume that a rooted forest with no edges consisting of $\ell$ roots of $F$ is also a valid subforest of $F$.For a valid subforest $F'=(F_1',F_2',\ldots,F_{\ell}')$ of $F$, we define $F-F':=(F_1^\ast,F_2^\ast,\ldots,F_{\ell}^\ast)$ to be a rooted forest-tuple in which for every $i\in [\ell]$, $F_i^\ast$ is a rooted forest such that $V(F_i^\ast)=V(F_i)$, $E(F_i^\ast)=E(F_i)\setminus E(F_i')$ and the root set of $F_i^\ast$ is $V(F_i')$. If $e$ is an edge incident to a root in $F_j$, then by $F-e$ we mean the rooted forest-tuple $F-F'$, where $F'_j$ has only one edge $e$ and the remaining forests of $F'$ have no edges.We say that $F$ is suitable if $3a(F) + k(F) - e(F) \leq 2$ or if $a(F)=1$ and $t(F)\ge 4$.We say that $F$ is $m$-suitable if it is a valid subforest of a suitable forest-tuple $F'$ with $e(F')=m$.We say that $F$ is simple if $t(F)\ge 4$ and either $a(F)=2$ and $k(F)=1$, or $a(F)=1$. Our main goal is to analyze whether Waiter can force a red copy of a forest $F$, based only on the sizes of the trees in $F$. We will show that if $F$ (or rather a forest-tuple made from $F$) is $m$-suitable, which roughly speaking means that every tree in $F$ has less than $m/3$ edges and $e(F)\le m$, then Waiter can force a red copy of $F$ on the board $K_{m+r}$, where $r=v(F)-e(F)$ is the number of roots. However, in some cases it is useful to have some additional properties under control, for example the number of blue edges between groups of red trees. This is why we do not simply divide a forest $F$ into trees, but rather into forests. Consequently, we will treat $F$ as a forest-tuple and consider sizes of the tuple forests instead of sizes of the trees. With a slight abuse of notation we will sometimes denote by $F$ also the rooted graph $F_1 \mathbin{\newline \hbox{\ialign{$\m@th$\cr\cup\cr\cdot\crcr}} } \displaylimits F_2 \mathbin{\newline \hbox{\ialign{$\m@th$\cr\cup\cr\cdot\crcr}} } \displaylimits \ldots \mathbin{\newline \hbox{\ialign{$\m@th$\cr\cup\cr\cdot\crcr}} } \displaylimits F_{\ell}$ corresponding to the rooted forest-tuple $F=(F_1,F_2,\dots,F_\ell)$. Before we proceed, let us focus on some properties of suitable and $m$-suitable rooted forest-tuples.

Key Result

Theorem 1.2

For every $\varepsilon\in\left(0,\frac{1}{3}\right)$ there exist positive constants $b$ and $n_0$ such that the following holds. Let $T_n$ be a tree on $n\geq n_0$ vertices with $\Delta(T_n)<\left(\frac{1}{3}-\varepsilon\right)n$. Then Waiter has a winning strategy in $\textrm{WC}(n,T_n)$. Furthermo

Theorems & Definitions (3)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4