Colorful subhypergraphs in Kneser hypergraphs

Using a Zq-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a uniform hypergraph).


Introduction 1.Motivations and results
A hypergraph is a pair H = (V (H), E(H)), where V (H) is a finite set and E(H) a family of subsets of V (H).The set V (H) is called the vertex set and the set E(H) is called the edge set.A graph is a hypergraph each edge of which is of cardinality two.A quniform hypergraph is a hypergraph each edge of which is of cardinality q.The notions of graphs and 2-uniform hypergraphs therefore coincide.If a hypergraph has its vertex set partitioned into subsets V 1 , . . ., V q so that each edge intersects each V i at exactly one vertex, then it is called a q-uniform q-partite hypergraph.The sets V 1 , . . ., V q are called the parts of the hypergraph.When q = 2, such a hypergraph is a graph and said to be bipartite.A q-uniform q-partite hypergraph is said to be complete if all possible edges exist.
A coloring of a hypergraph is a map c : V (H) → [t] for some positive integer t.A coloring is said to be proper if there is no monochromatic edge, i.e. no edge e with |c(e)| = 1.The chromatic number of such a hypergraph, denoted χ(H), is the minimal value of t for which a proper coloring exists.Given X ⊆ V (H), the hypergraph with vertex set X and with edge set {e ∈ E(H) : e ⊆ X} is the subhypergraph of H induced by X and is denoted H[X].
Given a hypergraph H = (V (H), E(H)), we define the Kneser graph KG 2 (H) by The "usual" Kneser graphs, which have been extensively studied -see [20,21] among many references, some of them being given elsewhere in the present paper -are the special cases H = ([n], [n]  k ) for some positive integers n and k with n 2k.We denote them KG 2 (n, k).The main result for "usual" Kneser graphs is Lovász's theorem [11].
The 2-colorability defect cd 2 (H) of a hypergraph H has been introduced by Dol'nikov [3] in 1988 for a generalization of Lovász's theorem.It is defined as the minimum number of vertices that must be removed from H so that the hypergraph induced by the remaining vertices is of chromatic number at most 2: Theorem (Dol'nikov theorem).Let H be a hypergraph and assume that ∅ is not an edge of H. Then χ(KG 2 (H)) cd 2 (H).
The following theorem proposed by Simonyi and Tardos in 2007 [19] generalizes Dol'nikov's theorem.The special case for "usual" Kneser graphs is due to Ky Fan [7].
Theorem (Simonyi-Tardos theorem).Let H be a hypergraph and assume that ∅ is not an edge of H. Let r = cd 2 (H).Then any proper coloring of KG 2 (H) with colors 1, . . ., t (t arbitrary) must contain a completely multicolored complete bipartite graph K r/2 , r/2 such that the r different colors occur alternating on the two parts of the bipartite graph with respect to their natural order.
the electronic journal of combinatorics 21(1) (2014), #P1.8 A Kneser hypergraph is thus the generalization of Kneser graphs obtained when the 2uniformity is replaced by the q-uniformity for an integer q 2. There are also "usual" Kneser hypergraphs, which are obtained with the same hypergraph H as for "usual" Kneser graphs, i.e.H = ([n], [n]  k ).They are denoted KG q (n, k).The main result for them is the following generalization of Lovász's theorem conjectured by Erdős and proved by Alon, Frankl, and Lovász [2].
There exists also a q-colorability defect cd q (H), introduced by Kříž, defined as the minimum number of vertices that must be removed from H so that the hypergraph induced by the remaining vertices is of chromatic number at most q: The following theorem, due to Kříž [9,10], generalizes Dol'nikov's theorem.It also generalizes the Alon-Frankl-Lovász theorem since cd q ([n], [n]  k ) = n − q(k − 1) and since again the inequality χ(KG q (n, k)) is the easy one.
Theorem (Kříž theorem).Let H be a hypergraph and assume that ∅ is not an edge of H. Then χ(KG q (H)) cd q (H) q − 1 for any integer q 2.
Our main result is the following extension of Simonyi-Tardos's theorem to Kneser hypergraphs.
Theorem 1.Let H be a hypergraph and assume that ∅ is not an edge of H. Let p be a prime number.Then any proper coloring c of KG p (H) with colors 1, . . ., t (t arbitrary) must contain a complete p-uniform p-partite hypergraph with parts U 1 , . . ., U p satisfying the following properties.
• It has cd p (H) vertices.
• The values of |U j | for j = 1, . . ., p differ by at most one.
• For any j, the vertices of U j get distinct colors.
We get that each U j is of cardinality cd p (H)/p or cd p (H)/p .Note that Theorem 1 implies directly Kříž's theorem when q is a prime number p: each color may appear at most p − 1 times within the vertices and there are cd p (H) vertices.There is a standard derivation of Kříž's theorem for any q from the prime case, see [22,23].Theorem 1 is a generalization of Simonyi-Tardos's theorem except for a slight loss: when p = 2, we do not recover the alternation of the colors between the two parts.
Whether Theorem 1 is true for non-prime p is an open question.

Local chromatic number and Kneser hypergraphs
In a graph G = (V, E), the closed neighborhood of a vertex u, denoted N [u], is the set {u} ∪ {v : uv ∈ E}.The local chromatic number of a graph G = (V, E), denoted χ (G), is the maximum number of colors appearing in the closed neighborhood of a vertex minimized over all proper colorings: where the minimum is taken over all proper colorings c of G.This number has been defined in 1986 by Erdős, Füredi, Hajnal, Komjáth, Rödl, and Seress [5].For Kneser graphs, we have the following theorem, which is a consequence of the Simonyi-Tardos theorem: any vertex of the part with r/2 vertices in the completely multicolored complete bipartite subgraph has at least r/2 + 1 colors in its closed neighborhhod (where r = cd 2 (H)).
Theorem (Simonyi-Tardos theorem for local chromatic number).Let H be a hypergraph and assume that ∅ is not an edge of Note that we can also see this theorem as a direct consequence of Theorem 1 in [18] (with the help of Theorem 1 in [13]).
We use the following natural definition for the local chromatic number χ (H) of a uniform hypergraph H = (V, E).For a subset X of V , we denote by N (X) the set of vertices v such that v is the sole vertex outside X for some edge in E: for any vertex v.The definition of the local chromatic number of a hypergraph is then: where the minimum is taken over all proper colorings c of H.When the hypergraph H is a graph, we get the usual notion of local chromatic number for graphs.
The following theorem is a consequence of Theorem 1 and generalizes the Simonyi-Tardos theorem for local chromatic number to Kneser hypergraphs.Theorem 2. Let H be a hypergraph and assume that ∅ is not an edge of H. Then for any prime number p.
the electronic journal of combinatorics 21(1) (2014), #P1.8 Proof.Denote cd p (H) by r.Let c be any proper coloring of KG p (H).Consider the complete p-uniform p-partite hypergraph G in KG p (H) whose existence is ensured by Theorem 1. Choose U j of cardinality r/p .If r/(p − 1) > r/p , then there is a vertex v of G not in U j whose color is distinct of all colors used in U j .Choose any edge e of G containing v and let u be the unique vertex of e ∩ U j .We have then |c Otherwise, r/(p − 1) = r/p , and for any edge e, we have |c(N [e \ {u}])| r/p = r/(p − 1) , with u being again the unique vertex of e ∩ U j .
As for Theorem 1, we do not know whether this theorem remains true for non-prime p.
3 Combinatorial topology and proof of the main result 3.1 Tools of combinatorial topology

Basic definitions
We use the cyclic and muliplicative group Z q = {ω j : j = 1, . . ., q} of the qth roots of unity.We emphasize that 0 is not considered as an element of Z q .For a vector X = (x 1 , . . ., x n ) ∈ (Z q ∪ {0}) n , we define X j to be the set {i ∈ [n] : We assume basic knowledges in algebraic topology, see the book by Munkres for instance for an introduction to this topic [17].A simplicial complex is said to be pure if all maximal simplices for inclusion have the same dimension.For K a simplicial complex, we denote by C(K) its chain complex.We always assume that the coefficients are taken in Z.

Special simplicial complexes
For a simplicial complex K, its first barycentric subdivision is denoted by sd(K).It is the simplicial complex whose vertices are the nonempty simplices of K and whose simplices are the collections of simplices of K that are pairwise comparable for ⊆ (these collections are usually called chains in the poset terminology, with a different meaning as the one used above in "chain complexes").
As a simplicial complex, Z q is seen as being 0-dimensional and with q vertices.Z * d q is the join of d copies of Z q .It is a pure simplicial complex of dimension d − 1.A vertex v taken in the µth copy of Z q in Z * d q is also written ( , µ) where ∈ Z q and µ ∈ [d].Sometimes, is called the sign of the vertex, and µ its absolute value.This latter quantity is denoted |v|.
We denote by σ q−1 q−2 the simplicial complex obtained from a (q − 1)-dimensional simplex and its faces by deleting the maximal face.It is hence a (q−2)-dimensional pseudomanifold homeomorphic to the (q − 2)-sphere.We also identify its vertices with Z q .A vertex of the simplicial complex σ q−1 q−2 * d is again denoted by ( , µ) where ∈ Z q and µ ∈ [d].For ∈ Z q and a simplex τ of σ p−1 p−2 * d , we denote by τ the set of all vertices of τ having as sign, i.e. τ := {(ω, µ) ∈ τ : ω = }.Note that if q is a prime number, Z q acts freely on σ q−1 q−2 .

Barycentric subdivision operator
Let K be a simplicial complex.There is a natural chain map sd # : C(K) → C(sd(K)) which, when evaluated on a d-simplex σ ∈ K, returns the sum of all d-simplices in sd(K) contained in σ, with the induced orientation."Contained" is understood according to the geometric interpretation of the barycentric subdivision.If K is a free Z q -simplicial complex, sd # is a Z q -equivariant map.

3.1.4
The Z q -Fan lemma The following lemma plays a central role in the proof of Theorem 1.It is proved (implicitely and in a more general version) in [8,14].
This ρ is an alternating simplex.
Proof.The proof is exactly the proof of Theorem 5.4 (p.415) of [8].The complex X in the statement of this Theorem 5.4 is our complex sd(Z * n q ), the dimension r is n − 1, and the generalized r-sphere (x i ) is any generalized (n − 1)-sphere of sd(Z * n q ) with x 0 reduced to a single point.The chain map h • is induced by our chain map λ # , instead of being induced by the chain map # of [8] (itself induced by the labeling ).It does not change the proof since h • only uses the fact that # is a Z q -equivariant chain map.In the statement of Theorem 5.4 of [8], α i is always a lower bound on the number of "alternating patterns" (i.e.simplices ρ as in the statement of the lemma) in # (x i ), even for odd i since the map f i in Theorem 5.4 of [8] is zero on non-alternating elements.Since α 0 = 1, we get that α i = 0 for all 0 i n − 1.
In particular, for q = 2, it gives the Ky Fan theorem [6] used for instance in [7,15,18] to derive properties of Kneser graphs.

Proof of the main result
Proof of Theorem 1.We first sketch some steps in the proof.We assume given a proper coloring c of KG p (H).With the help of the coloring c, we build a Z p -equivariant chain map ψ # : C(sd(Z * n p )) → C(Z * m p ), where m = n − cd p (H) + t(p − 1).We apply Lemma 3 to get the existence of some alternating simplex ρ in sd(Z * n p ).Using properties of ψ # (especially the fact that it is a composition of maps in which simplicial maps are involved), we show that this alternating simplex provides a complete p-uniform p-partite hypergraph in H with the required properties.
Let r = cd p (H).Following the ideas of [12,22], we define with m = n − r + t(p − 1).We choose a total ordering on the subsets of [n].This ordering is only used to get a clean definition of f .
, by definition of the colorability defect, at least one of the X j 's with j ∈ [p] contains an edge of H. Choose j ∈ [p] such that there is S ⊆ X j with S ∈ E(H).In case several S are possible, choose the maximal one according to the total ordering .Its defines F (X) := S and f (X) := (ω j , n − r + c(F (X))).The definition of τ is illustrated on Figures 1 and 2.
We exhibit now some properties of ρ and ρ .Since g is a simplicial map, we know that there is a permutation π and a sequence To ease the following discussion, we define The simplex τ which leads to the definition of g.
We clearly have 1 for all , .Since the element added to τ k to get τ k+1 is added to a τ k with minimum cardinality, we have 1 for all , .By induction we have in particular We can now conclude.Using the fact that f is simplicial, we get that ρ = {X 1 , . . ., X n } where the X i are signed vectors with |X i | = i and X 1 ⊆ • • • ⊆ X n .Moreover, we have the electronic journal of combinatorics 21(1) (2014), #P1.8 f ({X z+1 , . . ., X n }) = τ n .Each X i provides a vertex F (X i ) of KG p (H) for i = z + 1, . . ., n.For each j, define U j to be the set of such vertices F (X i ) such that the sign of f (X i ) is ω j .The U j are subsets of vertices of KG p (H).For two distinct j and j , if F (X i ) ∈ U j and F (X i ) ∈ U j , we have F (X i ) ∩ F (X i ) = ∅.Thus, the U j induce in KG p (H) a complete p-partite p-uniform hypergraph with n − z vertices.Equation (1) indicates that the cardinalities of the U j differ by at most one.Since the f (X i ) are all distinct, each U j has all its vertices of distinct colors.

Alternation number 4.1 Definition
Alishahi and Hajiabolhassan [1], going on with ideas introduced in [16], defined the qalternation number alt q (H) of a hypergraph H. Using this parameter, we can improve upon some theorems involving the q-colorability defect.The q-alternation number is defined as follows.
Let q and n be positive integers.An alterning sequence is a sequence s 1 , s 2 , . . ., s n of elements of Z q such that s i = s i+1 for all i = 1, . . ., n − 1.For a vector X = (x 1 , . . ., x n ) ∈ (Z q ∪ {0}) n and a permutation π ∈ S n , we denote alt π (X) the maximum length of an alternating subsequence of the sequence x π(1) , . . ., x π(n) .Note that by definition this subsequence has no zero element.
Let H = (V, E) be a hypergraph with n vertices.We identify V and [n].The qalternation number alt q (H) of a hypergraph H with n vertices is defined as: (2) Note that this number does not depend on the way V and [n] have been identified.

Improving the results with the alternation number
Alishahi and Hajiabolhassan improved the Kříž theorem by the following theorem.
It is an improvement since we have |V (H)| − alt q (H) cd q (H) as it can be easily checked.This inequality is often strict, see [1].Theorem 1 and Theorem 2 can be similarly improved with the alternation number.Let π be the permutation on which the minimum is attained in Equation (2).We replace r = cd p (H) by r = |V (H)| − alt p (H) in both proofs of Theorem 1 and Theorem 2, and we replace |X| in the definition of f by alt π (X) in the proof of Theorem 1.There are no other changes and we get the following theorems.
Theorem 4. Let H be a hypergraph and assume that ∅ is not an edge of H. Let p be a prime number.Then any proper coloring c of KG p (H) with colors 1, . . ., t (t arbitrary) must contain a complete p-uniform p-partite hypergraph with parts U 1 , . . ., U p satisfying the following properties.
• The values of |U j | for j = 1, . . ., p differ by at most one.
• For any j, the vertices of U j get distinct colors.The special case of Theorem 4 when p = 2 is proved in [1] in a slightly more general form.

Complexity
It remains unclear whether the alternation number, or a good upper bound of it, can be computed efficiently.However, we can note that given a hypergraph H, computing the alternation number for a fixed permutation is an NP-hard problem.Proposition 6.Given a hypergraph H, a permutation π, and a number q, computing max{alt π (X) : X ∈ (Z q ∪ {0}) n with E(H[X j ]) = ∅ for j = 1, . . ., q} is NP-hard.
the electronic journal of combinatorics 21(1) (2014), #P1.8 Proof.The proof consists in proving that the problem of finding a maximum independent set in a graph can be polynomially reduced to our problem for q = 2, π = id, and H being some special graph.
Let G be a graph.Define G to be a copy of G and consider the join H of G and G .The join of two graphs is the disjoint union of the two graphs plus all edges vw with v a vertex of G and w a vertex of G .We number the vertices of G arbitrarily with a bijection ρ : V → [|V |].It gives the following numbering for the vertices of H.In H, a vertex v receives number 2ρ(v) − 1 and its copy v receives the number 2ρ(v).Let n = 2|V |.As usual, we denote the maximum cardinality of an independent set of G by α(G).
By definition of the numbering, we have alt id (Y ) = 2|I| and thus max{alt id (X) : X ∈ (Z 2 ∪ {0}) n with E(H[X j ]) = ∅ for j = 1, 2} 2α(G) Conversely, any X = (x 1 , . . ., x n ) ∈ (Z 2 ∪ {0}) n with E(H[X j ]) = ∅ for j = 1, 2 gives an independent set I in G and another I in G : take a longest alternating subsequence in X and define the set I as the set of vertices v such that x 2ρ(v)−1 = 0 and the set I as the set of vertices v such that x 2ρ(v) = 0. We have alt id (X) = |I| + |I | because two components of X with distinct index parities cannot be of same sign: each vertex of G is the neighbor of each vertex of G .Thus

Figure 1 :
Figure 1: An example of a simplex τ ∈ M.

Theorem 5 .
Let H be a hypergraph and assume that ∅ is not an edge of H. Thenχ (KG p (H)) min |V (H)| − alt p (H) p + 1, |V (H)| − alt p (H) p − 1for any prime number p.