Abstract
In this paper, in view of Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol’nikovKříž bound. Next, we introduce multiple Kneser hypergraphs and we specify the chromatic number of some multiple Kneser hypergraphs. For a vector of positive integers and a partition of , the multiple Kneser hypergraph is a hypergraph with the vertex set
whose edge set is consist of any pairwise disjoint vertices. We determine the chromatic number of multiple Kneser hypergraphs provided that or for any , we have . In particular, one can see that if , , and , then . This gives a positive answer to a problem of Naserasr and Tardif [The chromatic covering number of a graph, Journal of Graph Theory, 51 (3): 199–204, (2006)].
A subset is almost stable if for any two distinct elements , we have . The almost stable Kneser hypergraph has all stable subsets of as the vertex set and every tuple of pairwise disjoint vertices forms an edge. Meunier [The chromatic number of almost stable Kneser hypergraphs. J. Combin. Theory Ser. A, 118(6):1820–1828, 2011] showed for any positive integer , . We extend this result to a large family of Schrijver hypergraphs. Finally, we present a colorfultype result which confirms the existence of a completely multicolored complete bipartite graph in any coloring of a graph.
Keywords: Chromatic Number, Kneser Hypergraph, Tucker Lemma, TuckerKy Fan’s Lemma.
Subject classification: 05C15
On Chromatic Number of Kneser Hypergraphs
Meysam Alishahi and Hossein Hajiabolhassan
Department of Mathematics
Shahrood University of Technology, Shahrood, Iran
Department of Applied Mathematics and Computer Science
Technical University of Denmark
DK2800 Lyngby, Denmark
Department of Mathematical Sciences
Shahid Beheshti University, G.C.
P.O. Box 1983963113, Tehran, Iran
1 Introduction
In this section, we setup some notations and terminologies. Hereafter, the symbol stands for the set . A hypergraph is an ordered pair , where (the vertex set) is a finite set and (the edge set) is a family of distinct nonempty subsets of . Throughout this paper, we suppose that for some positive integer . Assume that , where ’s are pairwise disjoint subsets of . The induced hypergraph has and as the vertex set and the edge set, respectively. If every edge of hypergraph has size , then it is called an uniform hypergraph. A coloring of a hypergraph is a mapping such that every edge is not monochromatic. The minimum such that there exists a coloring for hypergraph is termed its chromatic number, and is denoted by . If has an edge of size 1, then we define the chromatic number of to be infinite. For any hypergraph and positive integer , the Kneser hypergraph has as its vertex set and the edge set consisting of all tuples of pairwise disjoint edges of . It is known that for any graph , there exists a hypergraph such that is isomorphic to .
A subset is stable (reps. almost stable) if any two distinct elements of are at least “at distance apart” on the cycle (reps. path), that is, (reps. ) for distinct . Hereafter, for a subset , the symbols , , and stand for the set of all subsets of , the set of all stable subsets of , and the set of all almost stable subsets of , respectively. One can see that . Assume that , , and . Hereafter, for any positive integer , the hypergraphs , , and are denoted by , , and , respectively. Also, , , and , are denoted by , , and , respectively. Moreover, hypergraphs , , and are termed the Kneser hypergraph, the stable Kneser hypergraph, and the almost stable Kneser hypergraph, respectively. In 1955, Kneser [16] conjectured . Later, Lovász [19], in his fascinating paper, confirmed the conjecture using algebraic topology. Next, Erdős [9] presented an upper bound for the chromatic number of Kneser hypergraphs and conjectured the equality. In [2], this conjecture has been confirmed and it was shown . In view of result of Erdős [9], one can conclude that for any positive integer , we have . Also, Meunier [24] has shown that when , then . Finding a lower bound for chromatic number of hypergraphs has been studied in the literature, see [7, 17, 18, 28, 29, 34, 35]. As an interesting result, we have Dol’nikovKříž bound. For a hypergraph , the colorability defect of , say , is the minimum number of vertices which should be excluded such that the induced hypergraph on the remaining vertices has chromatic number at most .
Alon et al. [3] constructed ideals in which are not nonatomic but they have the Nikodým property by using stable Kneser hypergraphs. In this regard, they studied the chromatic number of stable Kneser hypergraph and presented the following conjecture.
Conjecture A
. [3] Let and be positive integers where and . We have
In [3], it has been shown that the aforementioned conjecture holds when is a power of . As an approach to Conjecture A, Meunier [24] showed that and he strengthened the above conjecture as follows.
Conjecture B
. [24] Let and be positive integers where and . We have
In [15], it has been shown that if and is sufficiently large, then for the aforementioned conjecture holds.
This paper is organized as follows. In Section 1, we set up notations and terminologies. Section 2 will be concerned with Tucker’s lemma and its generalizations. In Section , we introduce the alternation number of hypergraphs, and in view of Tucker lemma, we present a lower bound for chromatic number of hypergraphs. This lower bound improves the wellknow Dol’nikovKříž lower bound (Theoreme A). In fact, the alternation number of a hypergraph can be considered as a generalization of colorability defect of hypergraphs. Section 4 is devoted to multiple Kneser hypergraphs. In this section, we determine the chromatic number of some multiple Kneser hypergraphs. In particular, we present a generalization of a result of Alon et al. [2] about chromatic number of Kneser hypergraphs. In Section 5, we extend Meunier’s result and it is shown that for any positive integers and , if is an even integer or , then for Schrijver hypergraph , we have . In the last Section, in view of TuckerKy Fan’s Lemma, we prove a colorfultype result which confirms the existence of a completely multicolored complete bipartite graph in any coloring of a graph in terms of its alternation number.
2 Tucker’s Lemma and Its Generalizations
In this section, we present Tucker’s lemma and some of its generalizations. In fact, Tucker’s lemma is a combinatorial version of the BorsukUlam theorem with several interesting applications. For more about BursukUlam’s theorem and Tucker’s lemma, see [20].
Throughout this paper, for any positive integer , let be a set of size where . Moreover, when is a prime integer, we assume that is a cyclic group of order and generator , i.e., . In particular, when , we set . Consider a ground set such that and . Assume that is a sequence of . The subsequence () is said to be an alternating sequence if any two consecutive terms in this subsequence are different. We denote by the size of a longest alternating subsequence of nonzero terms in . For instance, if and , then .
One can consider as the set of all singed subsets of , that is, the family of all of disjoint subsets of . Precisely, for and , we define . Throughout of this paper, for any , we use these representations interchangeably, i.e., or . Assume . By , we mean for each . Note that if , then any alternating subsequence of is also an alternating subsequence of and therefore . Also, note that if the first nonzero term in is , then any alternating subsequence of of maximum length begins with and also, we can conclude that contains the smallest integer. For a permutation of , by , we denote the length of a longest alternating subsequence of nonzero signs in , i.e., . In this terminology, is the same as , where is the identity permutation. Now, we are in a position to introduce Tucker’s lemma.
Lemma A
. (Tucker’s lemma [31] ) Suppose that is a positive integer and . Also, assume that for any signed set , we have . Then, there exist two signed sets and such that and also .
There exist several interesting applications of Tucker’s lemma in combinatorics. Among them, one can consider a combinatorial proof for LovászKneser’s Theorem by Matousek [21]. Also, there are various generalizations of Tucker’s lemma. Next lemma is a combinatorial variant of Tucker Lemma proved and modified in [34] and [24], respectively.
Lemma B
. (Tucker Lemma) Suppose that and are nonnegative integers, where , , and is a prime number. Also, let
be a map satisfying the following properties:

is a equivariant map, that is, for each , we have ;

for all , if , then ;

for all , if , then the ’s are not pairwise distinct for .
Then .
Another interesting generalization of the BorsukUlam theorem is Ky Fan’s lemma [10]. This lemma has been used in some papers to study some coloring properties of graphs, see [5, 12] .
Lemma C
. (TuckerKy Fan’s lemma [10]) Assume that and are positive integers and satisfying the following properties:

for any , we have (a equivariant map)

there are no any two signed sets and such that and .
Then there are signed sets such that where . In particular .
3 An Improvement of Dol’nikovKříž Theorem
For a hypergraph , a permutation of (a linear ordering of ), and positive integers and , set to be the largest integer such that there exists an with and that the chromatic number of hypergraph is at most . Indeed, is the largest integer such that there exists an with and none of the ’s contain any member of . For instance, one can see that . Also, hereafter, is denoted by . Now, set . Also, is termed the alternation number of (with respect to ) and the first alternation number of is denoted by . In this terminology, one can see that if , then .
For a hypergraph , define to be the maximum size of a set such that the induced hypergraph on , is an colorable hypergraph. One can see that . In view of Theorem A, we know . In fact, Theorem A is an immediate consequence of this result. To see this, one can check that for any permutation of , we have . According to the definition of , there is an , such that none of ’s () contain any member of and that . Set . Obviously, is a proper coloring of . This implies that . On the other hand, and therefore . In this section, we show that
Assume that is a proper coloring of with colors where . For any subset , we define , if there is no where , then set . For any , define , i.e.,
Now, we are ready to improve Theorem A.
Lemma 1
. Assume that is a hypergraph, and is a prime number. For any positive integer where , we have
Proof.
Consider an arbitrary total ordering on . To prove the assertion, it is enough to show that for any , we have . Without loss of generality, we can suppose . Let be properly colored with colors . For any , we denote its color by . Set and .
Now, define a map as follows

If and , set , where is the index of set containing the smallest integer ( is then the first nonzero term in ).

If and , in view of definition of , the chromatic number of is at least . Set , where is a positive integer such that there is an where , , and is the biggest such a subset respect to the total ordering (note that ).
One can check that is a equivariant map from to .
Let . If , then the size of longest alternating subsequences of nonzero terms of and are the same. Therefore, the first nonzero terms of and are equal; and equivalently, .
Assume that such that . According to the definition of , for each , there are and such that and . This implies that . If , then is an edge in . But, this is a contradiction because is a proper coloring and .
Now, we can apply the Tucker Lemma and conclude that and so .
Lemma 2
. Suppose that and are positive integers, , and is a permutation of . Also, assume that for each , are disjoint subsets of . If we set
and
then .
Proof.
Without loss of generality, we can suppose . Also, let . If form an alternating subsequence of (), then the set is called the index set of this alternating subsequence. Choose an alternating subsequence of such that is the smallest integer in and that for each , there is a where . For each , assume that is a longest alternating subsequence of . Now, we present an alternating subsequence of . Construct such that for each , and have the same index set in . It is straightforward to check that is an alternating subsequence of and also, .
Here, we extend Lemma 1 to any uniform hypergraph ( is not necessarily prime) for the first alternation number.
Lemma 3
. Let and be positive integers where . Also, assume that for any hypergraph , and . For any hypergraph , we have .
Proof.
It is enough to show that for any , . Without loss of generality, we can suppose . Let . On the contrary, suppose
(1) 
Define the hypergraph as follows
Now, according to the assumption of theorem and the definition of , for each , we have
Consequently,
(2) 
Claim: .
Suppose, contrary to our claim, that and so .
By definition of , there is an
such that and none of ’s contain any member of .
In particular, none of them lie in .
Therefore, by the definition of , we have
It means Therefore, for each , there are disjoint sets , such that and none of them contain any member of . In particular, none of them contain any member of . Set
By Lemma 2,
Note that and thus,
Since does not contain any member of , we get which contradicts inequality (1). So we have proved the Claim.
By the assumption of theorem and the claim, we have
and so,
(3) 
Now, consider a proper coloring of . By inequality (2), in every , there exists a color which has been assigned to disjoint members of . Now, define such that is the maximum color which assigns to disjoint sets in . Now, according to inequality (3), there are sets which are disjoint and from receive the same color . Thus we have sets such that and also, they are disjoint and assigns them the same color which is a contradiction.
Theorem 1
. For any hypergraph and positive integer , we have
Theorem 1 in general is better than Dol’nikovKříž lower bound. Ziegler in [34, 35] showed that . Therefore, Dol’nikovKříž Theorem implies that . Although, one can easily see that and thus by Theorem 1,
It is easy to see that . Therefore, in view of Lemma 1 for , we have the next corollary.
Corollary 1
. [27] If and are positive integers where , then we have .
4 Multiple Kneser Graphs
Throughout this section, we assume that and are positive integers where and . Furthermore, suppose that is a partition of and is a positive integer vector where and for any , we have . The multiple Kneser hypergraph is a hypergraph with the vertex set