Sharp bounds for the chromatic number of random Kneser graphs

Abstract

Given positive integers $n⩾2k$, the Kneser graph $K{G}_{n,k}$ is a graph whose vertex set is the collection of all k-element subsets of the set $\{1,\dots ,n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of $K{G}_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of the random Kneser graph $K{G}_{n,k}(p)$, that is, the graph obtained from $K{G}_{n,k}$ by including each of the edges of $K{G}_{n,k}$ independently and with probability p.

We prove that, for any fixed $k⩾3$, $\chi (K{G}_{n,k}(1/2))=n-\mathrm{\Theta }(\sqrt[2k-2]{{\log }_{2}n})$, as well as $\chi (K{G}_{n,2}(1/2))=n-\mathrm{\Theta }(\sqrt[2]{{\log }_{2}n\cdot {\log }_2{\log }_{2}n})$. We also prove that, for $k⩾(1+\epsilon )\log \log n$, we have $\chi (K{G}_{n,k}(1/2))⩾n-2k-10$. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. The bound on k in the second result is also tight up to a constant. We also discuss an interesting connection to an extremal problem on embeddability of complexes.

Introduction

For positive integers $n,k$, where $n⩾2k$, the Kneser graph $K{G}_{n,k}=(V,E)$ is the graph, whose vertex set V is the collection of all k-element subsets of the set $[n]:=\{1,\dots ,n\}$, and E is the collection of the pairs of disjoint sets from V. This notion was introduced by Kneser [25], who showed that $\chi (K{G}_{n,k})⩽n-2k+2$. He conjectured that, in fact, equality holds in this inequality. This was proved by Lovász [29], who introduced the use of topological methods in combinatorics in that paper. One of the motivations for the studies in the present paper is to develop tools that could potentially help to obtain a ‘robust’ combinatorial proof of Kneser's conjecture.

We remark that independent sets in $K{G}_{n,k}$ are intersecting families, and it is a famous result of Erdős, Ko and Rado [15] that $\alpha (K{G}_{n,k})=\left(\begin{array}{c}n-1\\ k-1\end{array}\right)$.

The notion of the random Kneser graph $K{G}_{n,k}(p)$ was introduced in [6], [7]. For $0<p<1$, the graph $K{G}_{n,k}(p)$ is constructed by including each edge of $K{G}_{n,k}$ in $K{G}_{n,k}(p)$ independently with probability p. The authors of [8] studied the independence number of $K{G}_{n,k}(p)$. Later, their results were strengthened in [4], [9], [10]). Interestingly, the independence number of $K{G}_{n,k}(p)$ stays exactly the same as the independence number of $K{G}_{n,k}$ in many regimes. Independence numbers of random subgraphs of generalized Kneser graphs and related questions were studied in [5], [6], [7], [32], [33]. In [26], the second author proposed to study the chromatic number of $K{G}_{n,k}(p)$. He proved that in various regimes the chromatic number of the random Kneser graph is very close to that of the Kneser graph. In particular, he showed that for any constant k and p there exists a constant C, such that a.a.s. (asymptotically almost surely)²

$$\chi (K{G}_{n,k}(p))⩾n-C{n}^{\frac{3}{2k}}.$$

Moreover, he showed that the same holds for the random Schrijver graph (defined analogously based on Schrijver graphs, cf. [26]). A better a.a.s. bound $\chi (K{G}_{n,k}(p))⩾n-C{n}^{\frac{2+o(1)}{2k-1}}$ was next obtained by Alishahi and Hajiabolhassan [1]. In a follow-up paper, the second author [27] improved the inequality (1) to

$$\chi (K{G}_{n,k}(p))⩾n-C{(n\log n)}^{1/k}$$

for some $C=C(p,k)$. The main result of this paper is the following theorem, which, in particular, significantly improves upon the bounds (1) and (2) and settles the problem in the case of constant k.

Theorem 1

For any fixed $k⩾3$ and $n\to \infty$, we a.a.s. have

$$\chi (K{G}_{n,k}(1/2))=n-\mathit{\Theta }\left(\sqrt[2k-2]{{\log }_{2}n}\right).$$

For $k=2$ and $n\to \infty$ we a.a.s. have

$$\chi (K{G}_{n,k}(1/2))=n-\mathit{\Theta }\left(\sqrt[2]{{\log }_{2}n\cdot {\log }_2{\log }_{2}n}\right).$$

For clarity, all our results are stated and proved for $p=1/2$. However, they are easy to extend to any constant or slowly decreasing p. For $k=1$, $K{G}_{n,k}$ is just the complete graph $K_n$, and thus $K{G}_{n,1}(p)=G(n,p)$. Therefore, we a.a.s. have $\chi (K{G}_{n,1}(p))=\mathrm{\Theta }(\frac{n}{\log n})$ (see, e.g., [2]), that is, an analogue of Theorem 1 cannot hold. We note that a weaker version of Theorem 1 was announced in the short note due to the first author and Raigorodskii [23].

The papers [26], [1], [27] were also concerned with the following question: when does the chromatic number drop by at most an additive constant factor? The best results here are due to the second author [27], who proved the following a.a.s. bound for any fixed $l⩾2$ and some absolute constant $C=C(l)$:

$$\chi (K{G}_{n,k}(1/2))⩾n-2k+2-2l\mathit{\text{if}}k⩾C{(n\log n)}^{1/l}.$$

In this paper, we provide a major improvement of (3), replacing the polynomial dependence of k on n by doubly logarithmic.

Theorem 2

For any $\epsilon >0$ and $n\to \infty$ we a.a.s. have

$$\chi (K{G}_{n,k}(1/2))⩾n-2k-10\mathit{\text{if}}k⩾(1+\epsilon ){\log }_2{\log }_{2}n.$$

The bound on k is tight up to a constant: in [27, Section 5] it is proved that the bound $\chi (K{G}_{n,k}(1/2))⩾n-2k+2-2l$ cannot hold for any fixed l if $k<(\frac{1}{2}-\epsilon ){\log }_2{\log }_{2}n$.

Remark

In an earlier version of this paper, we showed the bound $\chi (K{G}_{n,k}(1/2))⩾n-2k+2-2l$ for $k⩾C{\log }^{\frac{1}{2l-3}}n$. We also related this problem to certain extremal properties of complexes. See Section 2.1 for the statement of the problem concerning simplicial complexes and the discussion section for the explanation of the relationship between the two problems. Since then, Kaiser and Stehlík [22] provided an important construction of a sparse subgraph of Kneser graphs, which lead to the present improvement. We discuss their construction in Section 3.3.1. We also note that the best possible constant in front of the double log is directly related to the value of ζ from Problem 7 in Section 2.1.

Generalizing the notion of a Kneser graph, Alon, Frankl and Lovász in [3] studied the Kneser hypergraph $K{G}_{n,k}^r$. The vertex set of the Kneser hypergraph is the same as that of the Kneser graph $K{G}_{n,k}$, and the set of edges is formed by the r-tuples of pairwise disjoint sets. In particular, verifying a conjecture of Erdős [14], they determined that $\chi (K{G}_{n,k}^r)=\lceil \frac{n-r(k-1)}{r-1}\rceil$. Determining the independence number of $K{G}_{n,k}^r$ is a much harder problem due to Erdős [13] (known under the name of the Erdős Matching Conjecture), which remains unresolved in full generality. The best known results in this direction were obtained in [16] and, more recently, in [20]. See also [12], [18], [19] for related stability results.

We can define the random Kneser hypergraph $K{G}_{n,k}^r(p)$ in a similar way. Studying the chromatic number of $K{G}_{n,k}^r(p)$ was proposed by the second author in [26]. First lower bounds were obtained in [1] and then they were significantly improved in [27]. In particular, it was shown that for fixed k, p and $r⩾3$ there exists a constant C, such that a.a.s.

$$\chi (K{G}_{n,k}^r(p))⩾\frac{n}{r-1}-C{\log }^{\frac{1}{\alpha }}n,$$

where $\alpha =r(k-1)-\frac{2r-1}{r-1}$. Note that $\chi (K{G}_{n,k}^r)=\frac{n}{r-1}-O(1)$ for fixed $k,r$. In the third theorem, we give an almost matching upper bound for (5).

Theorem 3

Let $k,r$ be fixed. Then there exists a constant C, such that for $n\to \infty$ a.a.s. we have

$$\chi (K{G}_{n,k}^r(1/2))⩽\frac{n}{r-1}-C{\log }^{\frac{1}{r(k-1)}}n.$$

Theorem 3 implies the upper bound from Theorem 1 for $k⩾3$ and will be proved in Section 4. The proof of the upper bound from Theorem 1 for $k=2$ is given after that, in Section 4.1.

We do not go into more historical details here, and refer the reader to [27] for a longer introduction to the subject and a more detailed comparison of the bounds in different regimes.

We note that, while the methods for studying Kneser graphs and hypergraphs in [26], [1] were topological, in [27] combinatorial methods relating the structure of $K{G}_{n,k}$ and $K{G}_{n,k+l}$ were used. In this paper we use (different) combinatorial and probabilistic methods, which are based on the analysis of the structure of independent sets in $K{G}_{n,k}(p)$. The proof of Theorem 2 both in the new and old versions uses only one topological statement as a black box: the fact that chromatic number of a certain graph ($K{G}_{n,k}$ or its subgraph $X{G}_{n,k}$ from [22]) is $n-2k+2$.

In the next section we discuss some problems related to colorings of Kneser graphs; in Section 3 we give the proofs of the lower bound for Theorem 1 and of Theorem 2. In Section 4 we prove Theorem 3. In Section 5 we conclude, state some open problems and discuss the relationship between some of the questions from Section 2.1 to colorings of random Kneser graphs.