Uniform Coloured Hypergraphs and Blocking Sets

Transcription

1 Applied Mathematical Sciences, Vol. 7, 2013, no. 91, HIKARI Ltd, Uniform Coloured Hypergraphs and Blocking Sets Mario Gionfriddo Dipartimento di Matematica e Informatica, Università di Catania Viale A.Doria 6, Catania, Italy gionfriddo@dmi.unict.it Lorenzo Milazzo Dipartimento di Matematica e Informatica, Università di Catania Viale A.Doria 6, Catania, Italy milazzo@dmi.unict.it Copyright c 2013 Mario Gionfriddo and Lorenzo Milazzo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper some properties of k-chromatic and strong k-chromatic hypergraphs are proved. Conditions for the possible existence of blocking sets in Steiner systems are established. Mathematics Subject Classification: 05C15; 05B05 Keywords: Colourings; Chromatic number; Blocking set 1 Introduction An hypergraph H of order n is a the pair (X, B), where X is a vertex set, with X = n, and B a family of nonempty subsets of X, called edges, such that E B E = X. The number of edges in B is denoted by m H. If all the edges of

2 4536 Mario Gionfriddo and Lorenzo Milazzo B have the same cardinality, then H is called uniform hypergraph of rank and the elements of B are also called blocks. A colouring f of H is a surjective mapping of X into a colour set C such that f(e) > 1, for every edge E of B with E > 1. The minimum number of colours for which there exists a colouring of H is called the chromatic number of H and it is denoted by χ H. When a colouring f uses k colours, then it is said to be a k-colouring of H. Each k-colouring defines a partition of X in k colour-classes X i, whose cardinalities will be denoted by n i, for 1 i k. A colouring f of H is said to be a strong colouring if f(e) = E, for every edge E of B. The corresponding chromatic number is said strong chromatic number of H and it is denoted by χ H. It follows that: for any hypergraph H: χ(h) χ H. Steiner systems are an important class of hypergraphs. A Steiner system S λ (h, k, v) is an hypergraph Σ = (X, B), uniform of rank k, having order v, such that every h-subset Y of X has degree λ (i.e. it is contained in exactly λ edges of B). A blocking set of Σ is a subset B of X such that both B and C X B are transversal: B E, C X B E, for every E B. It is evident that the existence of a blocking set in the hypergraphs H is equivalent to the 2-colouring of H. In the literature there are many results about Steiner systems Σ and about the possible existence in Σ of blocking sets. Interesting problems have been studied by the authors about hypergraphs in [13], Steiner systems [9,24], colourings [22,23], blocking sets [10,11,12,15], and also by other authors [6]. G-designs are a generalization of Steiner systems. Given a graph G, ag- design is a pair Σ = (X; B), where X is a finite non empty set and B is a family of graphs, all isomorphic to G, such that every pair of distinct elements of X is an edge of exactly λ graphs of B. ForG-designs there is an interesting literature about existence problems and also about colourings [1,5,8,14,16,18,21]. Let C be a set of colours and let X be any set of elements. A distribution of colours of C on X [briefly DC] is a surjection Δ: X C.If C = k, then Δ partitions the vertex set X in k colour-classes X i, for i =1, 2,..., k, whose cardinality will be denoted by n i. Let Δ be a DC of C on X. Let q,r be respectively quotient and remainder of the division of X by C. If Δ defines r colour-classes containing q +1 vertices and C - r colour-classes containing q vertices, then Δ is said to be an equidistribution of colours of C on X [briefly EC]. An equidistribution of colours will be indicate by E. Let be a positive integer and let Δ be a DC of C on X. Consider, if it is possible, the family B containing all the possible distinct subsets of X having cardinalities [-uples] for which Δ verifies the conditions to be a colouring.

3 Uniform coloured hypergraphs and blocking sets 4537 The pair H=(X, B) defines an uniform hypergraph of rank, which is called the hypergraph associated with Δ and it will be denoted by H(, Δ). It is possible to define also the hypergraphs strongly associated with Δ and it will be denoted by K(, Δ) and also K(, E), when it is the case of EC. The edges of K(, E) are formed by all the subsets of X having cardinalities and having no two vertices in a same class of Δ. Given a distribution of colours, it is possible that some hypergraph associated with it is not defined. For example, if X =5, C =2, = 4 and Δ is the DC which partitions X in two colour-classes with cardinalities n 1 =1 and n 2 = 4, then K(, Δ) is not defined. In this paper, we prove some necessary conditions for uniform hypergraphs having a given chromatic number and a given strong chromatic number. A consequent condition for Steiner systems devoid of blocking sets is given. We point out the problem to apply the main results found in this paper to determine the chromatic numbers of Steiner systems and of G-designs, especially if consider the combinatorial structures defined in [2,3,4,7,19,20,21]. Similar conditions can be found also for chromatic indices and can be applied to study the problems examined in [5,8,14,16,18]. In what follows, given two positive integer numbers k, h, we will indicate by Q(k, h),r(k, h) respectively the quotient and the rest of the division of k by h. 2 Preliminary results In this section we give some properties about distributions and equidistributions of colours. The proofs of the first two Lemmas can be obtained by simple calculations and they are omitted. Lemma 2.1 Let Δ : X Cbe a colour distribution, X = n, C = p. Let q = Q(n, p), r = R(n, p). It follows that: i) If Δ has a colour-class X i with n i <qvertices, then there exists a colourclass X j with n j q + 1 vertices. ii) If Δ has a colour-class X j with n j >q+ 1 vertices, then there exists a colour-class X j with n i q vertices. Lemma 2.2 A distribution of colours Δ is an equidistribution of colours if and only if X i X j 1, for every pair of colour-classes X i, X j. The next Lemma gives a technical result of combinatorial calculus. Lemma 2.3 If, s, t N and 1 < s<t 1, then

4 4538 Mario Gionfriddo and Lorenzo Milazzo ( ) t ( ) ( ) t 1 s +1 > ( ) s. Proof. - Induction on. For = 2, the inequality is true. Consider > 2 and suppose that the statement is true for -1. For every n, k N with k n we have that: ( ) ( ) n n n k +1 =. k k 1 k The statement follows from the following sequence of equalities and inequalities: = ( ) t > = ( ) t 1 = [( ) t 1 ( ) t t +1 1 ( ) t 1 t 1 ( )] ( ) t 1 t t > [( ) ( )] s +1 s s ( ) ( ) s +1 s +2 s s +1 = 1 1 ( ) s = ( ) s +1 = ( ) s. 3 Hypergraphs associated with distribution of colours In this section we prove some results about hypergraphs associated with classical colourings. The next Lemma provides an important tool. Lemma 3.1 Let Δ be a DC of C on X which defines two colour-classes X i and X j such that n j >n i +1.IfΔ is a DC obtained moving a vertex from X j to X i, then m H(,Δ) <m H(,Δ ).

5 Uniform coloured hypergraphs and blocking sets 4539 Proof. - In the edge-set B of H(, Δ) there are not monochromatic -uples of X. Ifn and p are respectively the cardinalities of X, C, and μ is the number of all the monochromatic -uples in X which are not contained in X i and X j, then - the number of all the -uples of X is ( n ) ; - the number of all the monochromatic -uples contained in X i with respect to Δ and Δ are ( n i ) ( and ni ) +1 ; - the number of all the monochromatic -uples contained in X j with respect to Δ and Δ are ( n j ) ( and nj ) 1. We have that: m H(,Δ) = m H(,Δ ) = ( ) n μ ( ) n μ ( ) ni ( ) nj, ( ) ( ni +1 nj 1 Setting s = n i and t = n j, the proof follows from Lemma 2.3. ). Theorem 3.1 If Δ and E are respectively a DC and an EC of C on X, then: m H(,Δ) m H(,E). Proof. - If Δ is an equidistribution of colours, then m H(,Δ) = m H(,E). If Δ is not an EC, then (Lemma 2.2) there exist at least two colour-classes of Δ, let X i and X j, having respectively cardinalities n i and n j, with n j >n i + 1. Then, it is possible to move a vertex from X j to X i, obtaining a new distribution of colours Δ 1. We can repeat the previous step until to obtain an equidistribution of colours Δ h. From Lemma 3.1 it follows that: m H(,Δ) <m H(,Δ1 ) <m H(,Δ2 ) < <m H(,Δh ) = m H(,E), where we have considered that two hypergraphs associated respectively with two ECs have the same number of edges. So, the statement is proved. Corollary 3.1 If Δ is a distribution of colours of C on X, then H(, Δ) has the maximum number of blocks if and only if Δ is an equidistribution of colours. Proof. - Suppose that m H(,D) m H(,Δ), for every distribution of colours D. If Δ is not an equidistribution of colours, then Δ defines two colour classes X i and X j such that n j >n i + 1 (Lemma 2.2). In this case, if we move a vertex

6 4540 Mario Gionfriddo and Lorenzo Milazzo from X j to X i, we obtain a new distribution of colours Δ with m H(,Δ) m H(,Δ ) (Lemma 3.1). Since this is a contradiction, Δ is necessarily an EC. Now, suppose that Δ is an equidistribution of colours. Then, from Theorem 3.1, for every distribution of colours D of C on X it is m H(,D) m H(,Δ). Theorem 3.2 Let E be an equidistribution of colours of C on X, with X = n and C = p. Ifq = Q(n, p) and r = R(n, p), then ( ) ( ) ( ) n q +1 q m H(,E) = r (p r). Proof. - The equidistribution E defines in X r colour-classes containing q + 1 vertices, and p - r colour-classes containing q vertices. The proof follows because there are not monochromatic -uples in the edge-set of m H(,E). Now, it is possible to determine an upper bound for the cardinality of the edge-set of an uniform hypergraph H having a given chromatic number. Corollary 3.2 Let H=(X,B) be an uniform hypergraph of rank and order n, having chromatic number χ H = χ. Ifq = Q(n, χ) and r = R(n, χ), then m H ( ) n ( q +1 ) r ( ) q (χ r). Proof. The proof follows from the previous Theorems. Theorem 3.3 If E, E are two equidistributions of colours respectively of C and of C on X, with X = n, C = p and C = p +1, then m H(,E) m H(,E ). Proof. - We can suppose that C Cand that c p+1 is the unique colour which is not in C, i.e. C = C {c p+1 }. Certainly, there exists a vertex x 0 X which belongs to a colour-class of E having cardinality more than one. Then, consider the distribution of colours of C, so defined: Δ(x) = { cp+1 if x = x 0 E(x) if x x 0.

7 Uniform coloured hypergraphs and blocking sets 4541 If μ, μ indicate the number of all the monochromatic -uples in X of E and Δ respectively, then, since μ μ, it follows that: m H(,E) = ( ) n μ ( ) n μ = m H(,Δ). Since Δ and E are two DC of C on X, then (Theorem 3.1) it follows that m H(,Δ) m H(,E ) and the Theorem is proved. 4 Hypergraphs strongly associated with distribution of colours In this section we prove some results about hypergraphs strongly associated with distributions of colours. Lemma 4.1 Let Δ be a distribution of colours of C on X having two colourclasses X i and X j, with n j >n i +1.IfΔ is a distribution of colours obtained from Δ moving a vertex from X j to X i, then m K(,Δ) <m K(,Δ ). Proof. - At first, observe that the edge-set of an hypergraph associated with a strong colouring contains only the -uples of X coloured with distinct colours. Let α i be the number of all the ( i)-uples coloured with i colours, NOT containing vertices of X i or X j, for i = 0, 1 and 2. Let Δ be the new distribution of colours obtained by moving a vertex from X j to X i.it follows: m K(,Δ) )=α 0 + α 1 n i + α 1 n j + α 2 n i n j, m K(,Δ ))=α 0 + α 1 (n i +1)+α 1 (n j 1) + α 2 (n i + 1)(n j 1), hence: m K(,Δ )) m K(,Δ) )=α 2 (n j n i 1) > 0, which proves the Lemma. The proofs of the next theorem and corollary are omitted. They can be obtained following the proofs of Theorem 3.1 and Corollary 3.1. Theorem 4.1 If Δ and E are respectively a distribution of colours and an equidistribution of colours of C on X, then m K(,Δ) ) m K(,Δ) ). Corollary 4.1 If Δ is a distribution of colours of C on X, then K(, Δ) has the maximum number of edges if and only if Δ is an equidistribution of colours.

8 4542 Mario Gionfriddo and Lorenzo Milazzo The following result permits to obtain an upper bound for the number of edges of an uniform hypergraph of rank having strong chromatic number χ. Theorem 4.2 Let E be an equidistribution of colours of C on X, with C = p and X = n. Ifq = Q(n, p) and r = R(n, p), then m K(,E) )= i=0 ( r i )( p r i ) (q +1) i q i. Proof. - Any -upla of K(, E) contains i vertices, for 0 i, belonging to i colour-classes of cardinality q + 1 and i vertices belonging to i colour-classes of cardinality q. It is possible to calculate the exact value of m K(,E), because there are ( r i) ways to fix i colour-classes with cardinality q + 1 and ( pr i) ways to fix i colour-classes with cardinality q. Corollary 4.2 Let H=(X,B) be an uniform hypergraph of rank. If q = Q(n, χ ) and r = R(n, χ ), where χ = χ H, then m H i=0 ( r i )( χ r i ) (q +1) i q i. Theorem 4.3 If E and E are two equidistributions of colours of of C on X, with E = p, E = p +1, then m K(,E) m K(,E ). Proof. - We can suppose that C Cand can indicate by c p+1 is the unique colour which is not in C. Let x 0 X be a vertex belonging in a colour-class of E, such that E > 1. We can consider the new distribution of colours Δ of C on X, so defined: { cp+1 if x = x Δ(x) = 0 E(x) if x x 0 Let ν be the number of all the -uples coloured with colours and not containing x 0, with respect to both DCs E and Δ, and let μ, μ be the number of all the -uples coloured with colours and containing the vertex x 0, with respect to the DC E and Δ respectively. Since μ μ, it follows: m K(,E) = ν + μ ν + μ = m K(,Δ). Since Δ and E are two DC of C, from Theorem 4.1 it follows m K(,Δ) m K(,E ), which proves the Theorem.

9 Uniform coloured hypergraphs and blocking sets About chromatic numbers The conditions found in the previous section can be useful to have indications about the chromatic numbers of uniform hypergraphs. Indeed, for every j {1, 2,..., n}, it is possible to determine the two integers q j = Q(n, j) and r j = R(n, j) and to define the following values: ( ) ( ) ( ) n qj +1 qj M j = r j (j r j ), N j = i=0 ( rj i )( ) j rj (q j +1) i q i j, i where: - M j is the number of edges of an hypergraphs associated with an equidistribution of colours Δ j : X C j, for C j = j (Theorem 3.2); - N j is the number of edges of hypergraphs strongly associated with the same equidistribution of colours Δ j (Theorem 4.2). About these parameters we have that: Theorem 5.1 For every M j and N j with j {1, 2,..., n} the following sequence of inequalities holds: M 1 M 2... M n, N 1 N 2... N n. Proof. - Let E j E j+1 be two equidistributions of colours respectively of C j and C j+1 on X. From Theorems 3.3 and 4.3 it follows: M j = m H(,E) m H(,E ) = M j+1 and the theorem is proved. N j = m K(,E) m K(,E ) = N j+1 It follows that there exist h,k {1,2,...,n-1} such that M h <m H M h+1, N k <m H N k+1. Since m H M χh (Theorem 2.6) and m H N χ H (Theorem 3.5), it follows: χ H h +1, χ H k +1. So, it is possible to examine which are the admissible values of the chromatic number and the strong chromatic number of uniform hypergraphs.

10 4544 Mario Gionfriddo and Lorenzo Milazzo 6 Hypergraphs uniform of rank =2, 3, 4 Directly from Corollary 3.2, it follows that: Theorem 6.1 If H=(X, B) is an hypergraph of order n, having chromatic number χ, uniform of rank =2, 3, 4, then: ( ) ( ) ( ) n q +1 q 1) m H r (χ r), for =2; ( ) ( ) ( ) n q +1 q 2) m H r (χ r), for =3; ( ) ( ) ( ) n q +1 q 3) m H r (χ r), for =4; where q = Q(n, χ), r = R(n, χ). These conditions, in the case of 2-colourings for hypergraphs of order n even, give the following: Theorem 6.2 If H=(X, B) is an hypergraph of order n even, having chromatic number 2, uniform of rank =2, 3, 4, then: 1) m H n2, for =2; 4 2) m H n2 (n2) 8, for =3; 3) m H n2 (n2)(7n22) 192, for =4. Proof. - From Theorem 6.1, for χ =2,n =2q, r = 0, by some calculations the statement follows. While, for n odd: Theorem 6.3 If H=(X, B) is an hypergraph of order n odd, having chromatic number 2, uniform of rank =2, 3, 4, then: 1) m H n2 1 4, for =2; 2) m H (n1)(n2 n2) 8, for =3; 3) m H (n1)(n3)(n+1)(7n15) 192, for =4.

11 Uniform coloured hypergraphs and blocking sets 4545 Proof. - Also in this case, from Theorem 6.1, for χ =2,n =2q +1, r =1,by some calculations the statement follows. 7 Steiner systems devoid of blocking sets Considering that existence of blocking sets and 2-colourings are two equivalent concepts in hypergraph theory and that the number of the edges in Steiner systems STS λ (n),sqs λ (n),s λ (2, 4,n) is respectively: n(n 1) n(n 1)(n 2) n(n 1) λ, λ, λ, it is possible to establish a connection between the index λ and the order n and to determine that values of λ, as function of n, for which the correspondent system does not admit blocking sets. Theorem 7.1 If Σ=(X, B) is a Steiner system S λ (2, 4,n) such that: i) λ> n(n 2)(7n 22), for n even, 16(n 1) or (n 3)(n + 1)(7n 15) ii) λ>, for n odd, 16n then there are no blocking sets in Σ. Proof. - Since in Σ it is: m = B = λ the condition i) follows from the inequality: λ n(n 1) 12 while the condition ii) follows from: λ n(n 1) 12 Hence, Σ is avoid of blocking sets. > n(n 1), 12 > n2 (n 2)(7n 22) ; 192 (n 1)(n 3)(n + 1)(7n 15). 192

12 4546 Mario Gionfriddo and Lorenzo Milazzo Theorem 7.2 If Σ=(X, B) is a Steiner system S λ (3, 4,n) such that: i) λ> n(7n 22), for n even, 8(n 1) or (n 3)(n + 1)(7n 15) ii) λ>, for n odd, 8n(n 2) then there are no blocking sets in Σ. Proof. - Since in Σ it is: n(n 1)(n 2) m = B = λ, 24 the condition i) follows from the inequality: n(n 1)(n 2) λ 24 while the condition ii) follows from: > n2 (n 2)(7n 22) ; 192 λ n(n 1)(n 2) 24 Hence, Σ is avoid of blocking sets. > (n 1)(n 3)(n + 1)(7n 15). 192 We have not considered the case of STS λ, because it is well-known that they are always devoid of blocking sets, except for n = 3 [11]. References [1] A.Amato, M.Gionfriddo, L-Milazzo, 2-regular equicolourings for P 4 designs, Discrete Math., 312 (2012), [2] L.Berardi, M.Gionfriddo, R.Rota, Perfect octagon quadrangle systems, Discrete Math., 310 (2010), [3] L.Berardi, M.Gionfriddo, R.Rota, Perfect octagon quadrangle systems - II, Discrete Math., 312 (2012), [4] L.Berardi, M.Gionfriddo, R.Rota, Balanced and strongly balanced P k - designs, Discrete Math., 312 (2012), [5] E.Billington, M.Gionfriddo, C.Lindner, The intersection problem for (K 4 e)- designs, J. Statist. Plann. Inference 58 (1997), 5 27.

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14 4548 Mario Gionfriddo and Lorenzo Milazzo [21] G.Lo Faro, A.Tripodi, The Doyen-Wilson theorem for kite systems, Discrete Math. 306 (2006), [22] L.Milazzo, The monochromatic block number, Discrete Math., (1997), [23] L.Milazzo, Z. Tuza, V.Voloshin, Strict colorings of Steiner triple and quadruple systems: A survey, Discrete Math., 261 (2003), [24] L.Milazzo, Z. Tuza, A class of Steiner systems S(2,4,v) with arcs of extremal size, Tatra Mt. Math. Publ., 36 (2007), Received: June 11, 2013