Coloring tournaments: from local to global

TL;DR

This paper demonstrates that tournaments with locally simple neighborhoods have bounded chromatic number, showing a significant simplification compared to general graphs, and answers a question posed by Berger et al.

Contribution

It establishes a bound on the chromatic number of tournaments based on local neighborhood properties, revealing a fundamental difference from undirected graphs.

Findings

Locally simple tournaments have bounded chromatic number

There exists a function f such that local chromatic bounds imply global bounds

Answers a question of Berger et al.

Abstract

The \emph{chromatic number} of a directed graph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class of $D$ induces an acyclic subdigraph. Thus, the chromatic number of a tournament $T$ is the minimum number of transitive subtournaments which cover the vertex set of $T$. We show in this paper that tournaments are significantly simpler than graphs with respect to coloring. Indeed, while undirected graphs can be altogether "locally simple" (every neighborhood is a stable set) and have large chromatic number, we show that locally simple tournaments are indeed simple. In particular, there is a function $f$ such that if the out-neighborhood of every vertex in a tournament $T$ has chromatic number at most $c$, then $T$ has chromatic number at most $f(c)$. This answers a question of Berger et al.

Equations

$${\gamma }_T(N^{+}[X])\le |X|,$$

$${\gamma }_T(Y)\le {\gamma }_T(X)+{\gamma }_T(Y\backslash X).$$

$$\gamma (V\setminus N^{+}[W])\ge \gamma (T)-\gamma (N^{+}[W])\ge K^{\prime }-|W|\ge K,$$

$$\gamma (N^{\prime })\ge \gamma (N^{+}(S))-\gamma (N^{+}(A))\ge K+\ell -|A|=K.$$

$${\ell }^{\prime }:=\ell +k(K+\ell +1)+\ell \left(\genfrac{}{}{0ex}{}{k(K+\ell +1)}{K+\ell +1}\right).$$

$$\vec{\chi }(N^{+}[v])\le \vec{\chi }(N^{+}(v))+\vec{\chi }(\{v\})\le t+1$$

$$\vec{\chi }(T)=\vec{\chi }(N^{+}[A])\le \sum _{v\in A}\vec{\chi }(N^{+}[v])\le (t+1)|A|=(t+1)\ell .$$

$$\vec{\chi }(T)=\vec{\chi }(N^{+}[D])\le \sum _{v\in D}\vec{\chi }(N^{+}[v])\le (t+1)|D|<(t+1)K.$$

Full text

**Keywords: ** chromatic number of tournaments, Erdős-Hajnal conjecture, digraph coloring

1 Introduction

A directed graph is said to be acyclic if it does not contain any directed cycles. Given a loopless digraph $D$, a $k$-coloring of $D$ is a coloring of each of the vertices of $D$ with one of the colors from the set $\{1,...,k\}$ such that each color class induces an acyclic subdigraph. The chromatic number $\vec{\chi}(D)$ of $D$ is the smallest number $k$ for which $D$ admits a $k$-coloring. This digraph invariant was introduced by Neumann-Lara [13], and naturally generalizes many results on the graph chromatic number (see, for example, [4], [9] [10], [11], [12]). In this paper, we study the chromatic number of a class of tournaments where the out-neighborhood of every vertex has bounded chromatic number.

A tournament is a loopless digraph such that for every pair of distinct vertices $u,v$, exactly one of $uv,vu$ is an arc. Given a tournament $T$, a subset $X$ of $V(T)$ is transitive if the subtournament of $T$ induced by $X$ contains no directed cycle. Thus, $\vec{\chi}(T)$ is the minimum $k$ such that $V(T)$ can be colored with $k$ colors where each color class is a transitive set. The coloring of tournaments has close relationship with the celebrated Erdős–Hajnal conjecture (cf. [1, 8]) and has been studied in [3, 5, 6, 2, 7].

Given $t\geq 1$, a tournament $T$ is $t$-local if for every vertex $v$, the subtournament of $T$ induced by the set of out-neighbors of $v$ has chromatic number at most $t$. The following conjecture was raised in [3] (Conjecture 2.6) and settled for $t=2$ in [7].

Conjecture 1**.**

There is a function $f$ such that every $t$-local tournament $T$ satisfies $\vec{\chi}(T)\leq f(t)$.

The goal of this note is to provide a proof of Conjecture 1 for all $t$.

Given a set $S\subset V(T)$, we say that $S$ is a dominating set of $T$ if every vertex in $V\setminus S$ has an in-neighbor in $S$. The dominating number $\gamma(T)$ of a tournament $T$ is the smallest number $k$ such that $T$ has a dominating set of size $k$. The main tool to prove Conjecture 1 is the following theorem, which seems more interesting than our original goal.

Theorem 2**.**

For every integer $k\geq 1$, there exist integers $K$ and $\ell$ such that every tournament $T$ with dominating number at least $K$ contains a subtournament on $\ell$ vertices and chromatic number at least $k$.

Roughly speaking, Theorem 2 asserts that if the dominating number of a tournament is sufficiently large, then it contains a bounded-size subtournament with large chromatic number. One may ask whether high dominating number is enough to force an induced copy of a specific (high chromatic number) subtournament. The following tournaments may be potential candidates. Let $S_{1}$ be the tournament with a single vertex. For every $i>1$, let $S_{i}$ be the tournament (with $2^{i}-1$ vertices) obtained by blowing up two vertices of an oriented triangle into two copies of $S_{i-1}$. It is easy to check that $\vec{\chi}(S_{i})\geq i$. The following problem is trivial for $i\leq 2$ and verified for $i=3$ in [7], while still open for all $i\geq 4$.

Problem 3**.**

For every integer $i\geq 1$, there exist $f(i)$ such that every tournament $T$ with dominating number at least $f(i)$ contains an isomorphic copy of $S_{i}$.

On another note, it is natural to ask whether Theorem 2 still holds with a weaker hypothesis. In particular, is it true that for every $k$, if the chromatic number of a tournament is huge, then it contains a bounded-size subtournament with chromatic number at least $k$? Unfortunately, the answer is negative for any $k\geq 3$. It is well-known that for any $\ell$, there is an undirected simple graph $G$ with arbitrarily high chromatic number and girth at least $\ell+1$. We fix an arbitrary enumeration of vertices of $G$ and create a tournament $T$ as follows: If $ij$ with $i<j$ is an edge of $G$ then $ij$ is an arc of $T$; otherwise, $ji$ is an arc of $T$. Then $T$ has arbitrarily high chromatic number while every subtournament of $T$ of size $\ell$ has chromatic number at most $2$. However, a similar question for dominating number is still open.

Problem 4**.**

For every integer $k\geq 1$, there exist integers $K$ and $\ell$ such that every tournament $T$ with dominating number at least $K$ contains a subtournament with $\ell$ vertices and dominating number at least $k$.

2 Proof of Conjecture 1

For every vertex $v$ in a tournament $T$, we denote by $N^{+}_{T}(v)$ the set of out-neighbors of $v$ in $T$. Given a subset $X$ of $V(T)$, let $N^{+}_{T}(X)$ denote the union of all $N^{+}_{T}(v)$, for $v\in X$, and denote by $N^{+}_{T}[X]:=X\cup N^{+}_{T}(X)$. For every subset $X$ of $V(T)$, let $\vec{\chi}_{T}(X)$ denote the chromatic number of the subtournament of $T$ induced by $X$.

Given a tournament $T$ and a subset $X$ of $V(T)$, we say a set $R\subseteq V(T)$ (not necessary disjoint from $X$) is a dominating set of $X$ in $T$ if every vertex in $X\backslash R$ has an in-neighbor in $R$. The dominating number $\gamma_{T}(X)$ of $X$ in $T$ is the smallest number $k$ such that $X$ has a dominating set of size $k$. When it is clear in the context, we omit the subscript $T$ in the notation.

Let $T$ be a tournament and $X,Y\subseteq V(T)$. The following inequalities are straightforward:

[TABLE]

and

[TABLE]

Let us restate Theorem 2.

Theorem 5**.**

For every integer $k\geq 1$, there exist integers $K$ and $\ell$ such that every tournament $T$ with $\gamma(T)\geq K$ contains a subtournament $A$ on $\ell$ vertices and $\vec{\chi}(A)\geq k$.

Proof.

We proceed by induction on $k$. The claim is trivial for $k=1$. For $k=2$, we can choose $K=2$ and $\ell=3$. Indeed, if a tournament $T$ satisfies $\gamma(T)\geq K=2$, then $T$ is not transitive and thus it contains an oriented triangle $A$ of size $\ell=3$ and $\vec{\chi}(A)\geq k=2$.

Assuming now that $(K,\ell)$ exists for $k$, we want to find $(K^{\prime},\ell^{\prime})$ for $k+1$. For this, we set $K^{\prime}:=k(K+\ell+1)+K$, and fix $\ell^{\prime}$ later. Let $T$ be a tournament such that $\gamma(T)\geq K^{\prime}$. Let $D$ be a dominating set of $T$ of minimum size. Consider a subset $W$ of $D$ of size $k(K+\ell+1)$. From (1) and (2) we have

[TABLE]

where $V$ is the vertex set of $T$. Thus by induction hypothesis applied to $k$, one can find a set $A\subseteq V\setminus N^{+}[W]$ such that $A$ has $\ell$ vertices and $\vec{\chi}(A)\geq k$. Note that by construction, $A\cap W=\emptyset$ and all arcs between $A$ and $W$ are directed from $A$ to $W$.

Consider now a subset $S$ of $W$ of size $K+\ell+1$. We claim that $\gamma(N^{+}(S))\geq K+\ell$. If not, we can choose a dominating set $S^{\prime}$ of $N^{+}(S)$ of size at most $K+\ell-1$. Note that $x$ dominates $S$ for any $x\in A$, and so $S^{\prime}\cup\{x\}$ dominates $N^{+}[S]$. Hence $(D\setminus S)\cup S^{\prime}\cup\{x\}$ would be a dominating set of $T$ of size less than $|D|$, which contradicts the minimality of $|D|$. Therefore $\gamma(N^{+}(S))\geq K+\ell$.

Let $N^{\prime}$ be the set of vertices $N^{+}(S)\setminus N^{+}(A)$. From (1) and (2) we have

[TABLE]

Thus by induction hypothesis applied to $k$, there is a subset $A_{S}$ of $N^{\prime}$ such that $|A_{S}|=\ell$ and $\vec{\chi}(A_{S})\geq k$. Note that by construction, $A_{S}\cap A=\emptyset$ and all arcs between $A_{S}$ and $A$ are directed from $A_{S}$ to $A$.

We now construct our subtournament of $T$ with chromatic number at least $k+1$. For this we consider the set of vertices $A\cup W$ to which we add the collection of $A_{S}$, for all subsets $S\subseteq W$ of size $K+\ell+1$. Call $A^{\prime}$ this new tournament and observe that its number of vertices is at most

[TABLE]

To conclude, it is sufficient to show that $\vec{\chi}(A^{\prime})\geq k+1$. Suppose not, and for contradiction, take a $k$-coloring of $A^{\prime}$. Since $|W|=k(K+\ell+1)$ there is a monochromatic set $S$ in $W$ of size $K+\ell+1$ (say, colored 1). Recall that we have all arcs from $A_{S}$ to $A$ and all arcs from $A$ to $S$, and note that since $\vec{\chi}(A)\geq k$ and $\vec{\chi}(A_{S})\geq k$, both $A$ and $A_{S}$ have a vertex of each of the $k$ colors. Hence there are $u\in A$ and $w\in A_{S}$ colored 1. Since $A_{S}\subseteq N^{+}(S)$, there is $v\in S$ such that $vw$ is an arc. We then obtain the monochromatic cycle $uvw$ of color 1, a contradiction. Thus, $\vec{\chi}(A^{\prime})\geq k+1$, completing the proof. ∎

We now show that Conjecture 1 is true.

Theorem 6**.**

There is a function $f$ such that every $t$-local tournament $T$ satisfies $\vec{\chi}(T)\leq f(t)$.

Proof.

Let $(K,\ell)$ satisfy Theorem 5 for $k:=t+1$. Let $T$ be a $t$-local tournament. Thus, if $\gamma(T)\geq K$ then $T$ contains a set $A$ of $\ell$ vertices and $\vec{\chi}(A)\geq t+1$. If a vertex $v\in V(T)\backslash A$ does not have an in-neighbor in $A$, then $A\subseteq N^{+}(v)$, and so $t+1\leq\vec{\chi}(A)\leq\vec{\chi}(N^{+}(v))\leq t$, a contradiction. Hence, $A$ is a dominating set of $T$. Note that

[TABLE]

for every $v\in V(T)$. Thus

[TABLE]

Otherwise, $\gamma(T)<K$. Let $D$ be a dominating set of $T$ with minimum size. Then

[TABLE]

Consequently, $t$-local tournaments have chromatic number at most f(t):=\max\big{(}(t+1)K,(t+1)\ell\big{)}. ∎

The implication of our result is that we are possibly missing a key-definition of what is a “large” (or “dense”) hypergraph (i.e., a set of subsets). It could be that for a suitable definition of “large” (for which “large” intersecting “large” would be “large”), we would obtain that for any tournament $T$ on vertex set $V$, the set of out-neighborhoods of vertices of $T$ is “large”, and in addition the set of subsets of vertices of a $K$-chromatic tournament inducing at least chromatic number $k$ is also “large”. Hence, if two large sets are intersecting in a non-empty way, one could find an out-neighborhood with chromatic number $k$.

If such a notion would exist, it should decorrelate the two large sets (out-neighborhoods and $k$-chromatic), and thus imply the following: If $T_{1},T_{2}$ are tournaments on the same set of vertices and $\vec{\chi}(T_{1})$ is huge, then there is a vertex $v$ such that $T_{1}$ induces on $N_{T_{2}}^{+}(v)$ a subtournament of large chromatic number. A very similar conjecture was proposed by Alex Scott and Paul Seymour.

Conjecture 7**.**

[14]** For every $k$, there exists $K$ such that if $T$ and $G$ are respectively a tournament and a graph on the same set of vertices with $G$ of chromatic number at least $K$, then there is a vertex $v$ such that $G$ induces on $N_{T}^{+}(v)$ a subgraph of $G$ of chromatic number at least $k$.