Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods

Abstract

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc uv in a tournament T is the set of vertices that form a directed triangle with arc uv. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erdős and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.

Proof of Theorem 3.1

Let D be a digraph with independence number $\alpha$, and let $X,Y \subseteq V(D)$. Then the following inequalities are straightforward.

$$\begin{aligned} \gamma (D[N^+[X]])\le & {} |X|,\nonumber \\ \gamma (D[Y])\le & {} \gamma (D[X]) + \gamma (D[Y\setminus X]). \end{aligned}$$

(A.1)

Theorem 3.1

There exist functions K and $\ell$ such that for every pair of integers $k, \alpha \ge 1$, every digraph D with independence number $\alpha$ and dominating number at least $K(\alpha ,k)$ contains a $(k, \ell (\alpha ,k))$-cluster.

Proof

Let $P(\alpha ,k)$ denote the statement of the theorem for $\alpha$ and k. Our goal is to prove $P(\alpha ,k)$ for all integers $\alpha , k \ge 1$. Let us assume that $P(\alpha -1,k)$ holds for all $k \ge 1$. The base case for this is P(1, k), which is proved in [9]. Now we fix $\alpha$ and we want to prove $P(\alpha ,k)$, which we will do by induction on k. The base case for this is $P(\alpha ,1)$, which is true since any digraph with independence number $\alpha$ and domination number at least 1 contains at least one vertex, which serves as a (1, 1)-cluster. To build intuition, we can also consider the next case, which is $P(\alpha ,2)$. This is true since any digraph with independence number $\alpha$ and domination number at least $\alpha +1$ contains a directed cycle of length at most $\ell (\alpha ,2) \le 2\alpha +1$, and this cycle requires two colors.⁵ Now we assume $P(\alpha ,k-1)$ (as well as $P(\alpha -1,k)$) and we want to prove $P(\alpha ,k)$.

We will follow the proof of Theorem 5 from [9]. Let us first prove a useful claim. Recall that $N^o(v)$ is the set of vertices that form non-edges with v.

Claim 1

If D does not contain a $(k, \ell (\alpha -1,k))$-cluster, then for any vertex $v \in V(D)$,

$$\begin{aligned} \gamma (D[N^o(v)]) \le K(\alpha -1,k). \end{aligned}$$

Proof

The digraph $D' = D[N^o(v)]$ has independence number $\alpha -1$. By the inductive hypothesis on $\alpha$, either $D'$ has a $(k, \ell (\alpha -1,k))$-cluster or $D'$ has domination number at most $K(\alpha -1,k)$. Thus, $\gamma (D[N^o(v)]) \le K(\alpha -1,k)$. $\square$

Let $D=(V,E)$ be a digraph with independence number $\alpha$ such that $\gamma (D) \ge K(\alpha ,k)$, and let B be a minimum dominating set of D. We will assume that D does not contain a $(k, \ell (\alpha -1,k))$-cluster, since otherwise, we would be done. Fix

$$\begin{aligned} K(\alpha ,k)= & {} k (K(\alpha -1,k) + 1)(K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1)\\{} & {} +K(\alpha ,k-1). \end{aligned}$$

Consider a subset W of B, where

$$\begin{aligned} |W| = k(K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1). \end{aligned}$$

From (A.1) and Claim 1, we have

$$\begin{aligned} \gamma (D[V \setminus (N^+[W] \cup N^o(W))])\ge & {} \gamma (D) - \gamma (D[N^+[W]]) - \gamma (D[N^o(W)])\\ {}\ge & {} \gamma (D) -|W| - |W|(K(\alpha -1,k)\\\ge & {} K(\alpha ,k)-|W|(K(\alpha -1,k)+1)\\ {}\ge & {} K(\alpha ,k-1). \end{aligned}$$

By applying the induction hypothesis, the digraph $D[V\setminus {(N^+[W] \cup N^o(W))}]$ contains a $(k-1, \ell (\alpha ,k-1))$-cluster. Call this vertex set A. Note that by construction, $A \cap W = \emptyset$ and A is complete towards W. Now consider a subset S of W where

$$\begin{aligned} |S| = K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1. \end{aligned}$$

We claim that

$$\begin{aligned} \gamma (D[N^+(S)]) \ge K(\alpha ,k-1) + \ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1). \end{aligned}$$

(A.2)

If not, we can choose a dominating set $S'$ of $N^+(S)$, where

$$\begin{aligned} |S'| \le K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)-1. \end{aligned}$$

Note that x dominates S for any $x \in A$, and so $S' \cup \{x\}$ dominates $N^+[S]$. Hence $(B\setminus S) \cup S' \cup \{x\}$ would be a dominating set of D of size less than |B| which contradicts the minimality of B. We therefore conclude that Inequality (A.2) holds.

Let $N' = N^+(S) \setminus (N^+(A) \cup N^o(A))$. From Claims A.1 and 1 we have

$$\begin{aligned} \gamma (D[N'])\ge & {} \gamma (D[N^+(S)]) - \gamma (D[N^+(A)]) - \gamma (D[N^o(A)])\\\ge & {} K(\alpha ,k-1) {+}\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1) {-} |A|(K(\alpha -1,k)+1)\\= & {} K(\alpha ,k-1). \end{aligned}$$

Thus, by the induction hypothesis on k, there is a subset $A_s \subseteq N'$ that forms a $(k-1, \ell (\alpha , k-1))$-cluster. By construction, $A_S \cap A = \emptyset$ and $A_S$ is complete towards A.

We now construct our subdigraph of D with chromatic number at least k. We consider the set of vertices $A \cup W$ to which we add the collection $A_S$, for all subsets $S \subseteq W$ of size $K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1$. Call $A'$ this new vertex set and observe that its size is at most

$$\begin{aligned} |A'| \le |A| + |W| + |A_S| {|W| \atopwithdelims ()|S|}. \end{aligned}$$

So we have

$$\begin{aligned} \ell (\alpha ,k) ={} & {} \ell (\alpha ,k-1) + k(K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1) \\{} & {} + \ell (\alpha ,k-1)\left( {\begin{array}{c}k(K(\alpha ,k-1) + \ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1)\\ K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1\end{array}}\right) . \end{aligned}$$

To conclude, it is sufficient to show that $\chi (A') \ge k$. Suppose not, and for contradiction, take a $(k-1)$-coloring of $A'$. Since $|W| = k(K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1)$ there is a monochromatic set S in W of size $K(\alpha ,k-1)+\ell (\alpha ,k-1)\cdot (K(\alpha -1,k)+1)+1$ (say, colored 1). Recall that $A_S$ is complete to A, and A is complete to S, and note that since $\chi (A) \ge k-1$ and $\chi (A_S) \ge k-1$, both A and $A_S$ have a vertex of each of the $k-1$ 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 (v, w) is an arc of D. We then obtain the monochromatic triangle (u, v, w) of color 1, a contradiction. Thus, $\vec {\chi }(D[A']) \ge k$ implying that $A'$ is a $(k,\ell (\alpha ,k))$-cluster in D completing the induction on k.

Since this induction proves the statement $P(\alpha ,k)$ holds for any k, it proves the inductive hypothesis for $\alpha$. Then, by induction on $\alpha$ we have proven that the theorem is true for any pair of integers $\alpha , k$. $\square$