Equiblocking Sets in K 2,2 -Designs and in K 3,3 -Designs

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1 Applied Mathematical Sciences, Vol. 7, 013, no. 91, HIKARI Ltd, Equiblocking Sets in K, -Designs and in K 3,3 -Designs Mario Gionfriddo Dipartimento di Matematica e Informatica, Università di Catania Viale A.Doria 6, 9515 Catania, Italy gionfriddo@dmi.unict.it Lorenzo Milazzo Dipartimento di Matematica e Informatica, Università di Catania Viale A.Doria 6, 9515 Catania, Italy milazzo@dmi.unict.it Copyright c 013 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 There are many papers in the literature about blocking sets in hypergraphs, including Steiner systems (see references). In this paper we study the existence of equi-blocking sets in K, -designs and in K 3,3 - designs, determining the spectrum. Further we determine K, -designs and K 3,3 -designs having blocking sets of all the possible cardinalities. Mathematics Subject Classification: 05B05; 05C15 Keywords: Equiblocking set; trasversal; K p,p design 1 Introduction Let K v be the complete undirected graph defined on the vertex set X. Let G be a subgraph of K v. A G-decomposition of K v, also called a G-design of order v, is a pair Σ = (X, B), where B is a partition of the edge set of K v into

2 4550 Mario Gionfriddo and Lorenzo Milazzo classes generating graphs all isomorphic to G. The classes of the partition B are said to be the blocks of Σ. An uniform hypergraph of rank n and order v can be considered a G-design whose blocks are all complete subgraphs K n of K v. Using hypergraph terminology, if Σ = (X, B) isag-design of order v, atransversal T of Σ is a subset of X intersecting every block of Σ. The transversal number of Σ is the minimum number τ(σ) = τ for which there exists a transversal of Σ having cardinality τ. A blocking set B of Σ is a subset of X such that both B and C X (B) are transversals. An equi-blocking set of Σ is a blocking set having cardinality v or v. It is immediate to see that the existence of blocking sets in a system is equivalent to the -colourings of the vertices of that system [1,9,10,11]. The problem to determine the existence of possible blocking sets in Steiner systems and G-designs has been studied by many authors [5,9,10,1]. There are many classes of G-designs for which the problem of the determination of blocking sets is still open [,3,4,6,7] and many problems could be examined. Among them, we point out the study of blocking sets in systems have some blocks in common [8]. In this paper we study the existence of equi-blocking sets in K, -designs and in K 3,3 -designs, determining the spectrum. Further we determine K, - designs and K 3,3 -designs having blocking sets of all the possible cardinalities. In what follows, we will indicate by [(x 1,x 3,x 5 ), (x,x 4,x 6 )] the complete bipartite graph K 3,3 =(X, Y ; E), defined on the two stable vertex-sets X = {x 1,x 3,x 5 } and Y = {x,x 4,x 6 }, while the complete bipartite graph K, will be indicated by [x 1,x,x 3,x 4 ], being a cycle C 4 of length four with edges {x i,x i+1 }, for i Z 4. It is well known that: 1) in the complete bipartite graph K n,n, every vertex has degree n; ) in a K n,n -design of order v, the number b of blocks and the degree r of the vertices are respectively: b = v(v 1)/n, r =(v 1)/n; 3) there exists a K 3,3 -design of order v if and only if v 1, mod9,v 19; 4) there exists a K, -design of order v if and only if v 1, mod8,v 9. Transversals in G-designs The following theorem gives a necessary condition for the existence of blocking sets in G-designs. Given a graph G =(V,E), ϑ indicates the minimum degree of a vertex x V. Theorem.1 : Let Σ=(X, B) be a G-design of order v, ϑ the minimum degree in G, T a transversal of Σ with T = p. If p ϑ v 1, then (1) ( ) p + p [ v 1 ϑ (p 1)] B.

3 Equiblocking sets in K, -designs and in K 3,3 -designs 4551 Proof. Let Σ = (X, B) beag-design of order v. Let P be a non-empty subset of X such that: P = p, p ϑ<v 1, where ϑ is the minimum degree of a vertex x in G. Let M(P ) be the maximum number of blocks of Σ which can be intersect P. Considering that every pair of distinct elements of X belongs to exactly one block, there are at most ( ) p blocks containing the pairs of P. Further, since every vertex of X forms v 1 pairs with all the other vertices of Σ and its minimum degree in G is ϑ, it follows that for every vertex x X there are at most other [v ϑ(p 1) 1] blocks containing it. From which: ( ) p M(P ) + p [ v 1 (p 1)]. ϑ At this point, we can conclude that if T is a transversal of Σ, then M(P ) B. Hence: ( ) p + p [ v 1 (p 1)] B. ϑ In the cases of K, -designs and K 3,3 -designs, we have the following results. Theorem. : If Σ=(X, B) is a K u,u -design of order v, for u =, 3, and B is a blocking set of Σ of cardinality p, then: p ( u + 1)k +1 8k +1. Proof. Let Σ = (X, B) beak u,u -design of order v, for u =, 3, and let B be a blocking set of Σ of cardinality p. For u =, since v =8k + 1, from (1) of Theorem.1 it follows: from which: p (8k +1)p +k(8k +1) 0, p 8k +1 8k +1. For u = 3, since v =9k +1,k, from (1) of Theorem.1 it follows: p (6k +1)p + k(9k +1) 0, from which: p 6k +1 8k +1.

4 455 Mario Gionfriddo and Lorenzo Milazzo In what follows, we will indicate by p min the minimum positive integer verifying the inequality () and by B(Σ) the set of all integers p for which there exists in Σ a blocking set of cardinality p. We will see also that the inequality (1) is the best possible, in the sense that there are G-designs with blocking sets of cardinality p min and that there are G-designs Σ with blocking sets having cardinality all the possible values p B(Σ). Finally, we determine the spectrum for equi-blocking sets in K, -designs and in K 3,3 -designs. 3 All blocking sets in K, -designs and K 3,3 - designs of small order In this section we determine all the possible cardinalities for blocking sets in K, -designs of order 9,17 and in K 3.3 -designs of order 19, K, -designs of order 9,17 Theorem 3.1 : If Σ = (X, B) is a K, -design of order 9, then B(Σ) = {3, 4, 5, 6}. Proof. Let Σ be a K, -design of order 9. From Theorem.1, if B is a blocking sets of Σ such that B = p, then: p =3, 4. But the existence of blocking sets of cardinality p implies the existence of blocking sets of cardinality v p. Therefore: B(Σ) = {3, 4, 5, 6}. Theorem 3. : If Σ = (X, B) is a K, -design of order 17, then B(Σ) = {7, 8, 9, 10}. Proof. Let Σ be a K, -design of order 17. From Theorem.1, if B is a blocking sets of Σ such that B = p, then: p =5, 6. Therefore: B(Σ) = {7, 8, 9, 10}, for the same remark of the previous Theorem. Theorem 3.3 : There exist K, -designs of order v =9having blocking sets of all the admissible cardinalities. Proof. Let X = Z 9 and let Σ = (X, B) be the K, -design of order v =9 whose blocks are the translates of the base-block: [0,1,5,]. This means that the blocks can be obtained from [(i, i +1,i+5,i+ ] for every i Z 9. We can verify that T 1 ={0, 3, 7} is a blocking set of Σ and this implies that also C X (T 1 )={1,, 4, 5, 6, 8} is a blocking set of Σ. Further, also T = {0, 1, 3, 7} and C X (T ) are blocking sets of Σ.

5 Equiblocking sets in K, -designs and in K 3,3 -designs 4553 Theorem 3.4 : There exist K, -designs of order v =17having blocking sets of all the admissible cardinalities. Proof. Let X = Z 17 and let Σ = (X, B) be the K, -design of order v =17 whose blocks are the translates of the base-blocks: [0,1,9,],[0,3,9,4]. This means that the blocks can be obtained from [(i, i +1,i+9,i+ 4] and [(i, i + 3,i+9,i+4], for every i Z 9. We can verify that T 1 ={1, 3, 6, 8, 10, 13, 15} and T ={0, 1,, 3, 4, 5, 6, 7} are two blocking sets of Σ of cardinality 7,8 respectively. Considering C X (T 1 ) and C X (T ) the statement follows. -3. K 3,3 -designs of order 19,8 Theorem 3.5 : If Σ=(X, B) is a K 3,3 -design of order 19, then {5, 6, 13, 14} B(Σ). Proof. Let Σ be a K 3,3 -design of order 19. From Theorem.1, if B is a blocking sets of Σ such that B = p, then: p =5, 6. Therefore: {5, 6, 13, 14} B(Σ). Theorem 3.6 : If Σ=(X, B) is a K 3,3 -design of order 8, then {7, 8, 9, 19, 0, 1} B(Σ). Proof. Let Σ be a K 3,3 -design of order 8. From Theorem.1, if B is a blocking sets of Σ such that B = p, then: p =7, 8, 9. Therefore: {7, 8, 9, 19, 0, 1} B(Σ). Theorem 3.7 : There exist K 3,3 -designs of order v =19having blocking sets of all the admissible cardinalities. Proof. Let X = Z 19 and let Σ = (X, B) be the K 3,3 -design of order v =19 whose blocks are the translates of the following base-blocks: [(0,9,10),(4,7,18)]. This means that the blocks can be obtained from [(i, i +9,i+ 10), (i +4,i+ 7,i+ 18)] for every i Z 19. We can verify that B={0, 1,, 3, 4, 5, 6, 7, 8} is a blocking set of Σ and this implies that also C X (B) = {9, 10,..., 17} is a blocking set. Finally, since B = 9 and C X (B) = 10, it follows that B and C X (B) are both equi-blocking sets. Theorem 3.8 : There exist K 3,3 -designs of order v =8having blocking sets of all the admissible cardinalities.

6 4554 Mario Gionfriddo and Lorenzo Milazzo Proof. Let X = Z 7, Y = {A, B, C, D}. IfB is the family of K 3,3 s defined in X Y as follows (where we indicate a pair (x, y) byx y ): [(i A, (i +1) B,i D ), ((i +) A,i B, (i +1) C )], [(i B, (i +1) C, (i +1) D ), ((i +) A, (i +5) B, (i +4) C )], [(i A,i B, (i +6) C ), ((i +4) A, (i +5) C, (i +4) D )], [(i A, (i +1) B, (i +6) D ), ((i +4) B, (i +3) C,i D )], [(i A, (i +1) B,i D ), ((i +6) A,i C, (i +) D )], [(i A, (i +) C, (i +6) D ), ((i +1) B, (i +4) C, (i +3) D )], for every i Z 7, then we can verify that Σ = (X Y,B) isak 3,3 -design of order v = 8. We can observe that in every block there are vertices belonging to X {A}, vertices belonging to X {B}, and also to X {C}, and finally to X {D}. This implies that X {A} is a blocking set of Σ and it is X {A} = 7 [the same thing is verified by X {B},X {C},X {D}]. Further, for the same reason, it follows that, for every M B X {B}, M B, the set X {B} M B is a blocking set of Σ. This means that in Σ there are blocking sets of size 7, 8,..., 14 and also, by their complementary sets, of cardinality 15,..., 1. 4 Equiblocking sets in K, and K 3,3 designs In this last section we determine the spectrum of K, -designs and K 3,3 -designs having equi-blocking sets. -4.1) Equiblocking sets in K, -designs Theorem 4.1 : If Σ=(X, B) is a K, -design of order v =8k +1 having a blocking set of cardinality p, then there exists a K, -design of order v =8k +9 having a blocking set of cardinality p +4. Proof. Let Σ 1 =(X 1, B 1 )beak, -design of order 8k + 1 having a blocking set B 1 of cardinality p and let Σ =(X, B )beak, -design of order 9 having a blocking set B of cardinality 4 (it there exists, see Theorem 3.1). Further, let X 1 = {0,a 1,a,..., a 8k }, X = Z 9, X 1 X = {0}, B = {1,, 3, 4}. Finally, consider the family F of K, s defined in X 1 X {0} as follows:

7 Equiblocking sets in K, -designs and in K 3,3 -designs 4555 [a i, 1,a i+k+1, 5], [a i,,a i+1, 6], [a i, 3,a i+1, 7], [a i, 4,a i+1, 8], for every i =1,,..., k. If X = X 1 X, B = B 1 B F, it is possible to verify that Σ = (X, B) is a K, -design of order v =8k + 8. Further, if B = B 1 B, B is a blocking set of Σ such that B = B 1 + B = p + 4. Observe that B 1 is a blocking set of Σ 1, B a blocking set of Σ and that every block of F contains at least a vertex of B, belonging to B 1, and at least a vertex of C X B, belonging to C X B 1. Theorem 4. : For every v = 8k +1, there exist K, -designs of order v having equi-blocking sets. Proof. For v =9, 17, see Theorem 3.1 and 3.. Let v 17, v =8k +1. If ϱ =(v 1)/, from Theorem 5.1 it is possible to construct a K 3,3 -design of order v = v + 8, having blocking sets of cardinality ϱ + 4. Since: ϱ +4= v 1 +4= v 1 = ϱ, the statement is proved. -4.) Equiblocking sets in K 3,3 -designs Theorem 4.3 : For every v =9k +1, k, there exist K 3,3 -designs of order v having equi-blocking sets. Proof. For v =19, 8, see Theorem 3.7 and 3.8. Let Σ 1 =(X 1, C 1 )bea K 3,3 -design of order v =9k +1,k N, k, defined in X 1 = Z 9k+1, having an equi-blocking set B 1, such that B 1 = 9k. Further let Σ =(X, C )be a K 3,3 -design of order v = 19, defined in X = {0,a 1,a,..., a 9,b 1,b,..., b 9, }, where X 1 X = {0}, having equi-blocking set B = {a 1,a,..., a 9 }. If F is a partition of X 1 {0} in triples: let P = {{1,, 3}, {4, 5, 6},..., {3h +1, 3h +, 3h +3},...{9k, 9k 1, 9k}}, then define a family F of K 3,3 s as follows: [(1,, 3), (a 1,a,b 1 )], [(4, 5, 6), (a 1,a,b 1 )],..., [(9k, 9k 1, 9k), (a 1,a,b 1 )], [(1,, 3), (a 3,a 4,b )], [(4, 5, 6), (a 3,a 4,b )],..., [(9k, 9k 1, 9k), (a 3,a 4,b )], [(1,, 3), (a 5,a 6,b 3 )], [(4, 5, 6), (a 5,a 6,b 3 )],..., [(9k, 9k 1, 9k), (a 5,a 6,b 3 )],

8 4556 Mario Gionfriddo and Lorenzo Milazzo [(1,, 3), (a 7,b 4,b 5 )], [(4, 5, 6), (a 7,b 4,b 5 )],..., [(9k, 9k 1, 9k), (a 7,b 4,b 5 )], [(1,, 3), (a 8,b 6,b 7 )], [(4, 5, 6), (a 8,b 6,b 7 )],..., [(9k, 9k 1, 9k), (a 8,b 6,b 7 )], [(1,, 3), (a 9,b 8,b 9 )], [(4, 5, 6), (a 9,b 8,b 9 )],..., [(9k, 9k 1, 9k), (a 9,b 8,b 9 )]. If X = X 1 X and C = C 1 C 1 F, then Σ =(X, C) isak 3,3 -design of order v =9k Further, we can verify that B = B 1 B is a blocking set of Σ and B = B 1 + B = 9k 9(k+) +9 =. Therefore, B is an equi-blocking set of Σ. References [1] A.Amato, M.Gionfriddo, L.Milazzo, -regular equicolourings for P 4 - designs, Discrete Math., 31 (01), [] L.Berardi, M.Gionfriddo, R.Rota, Perfect octagon quadrangle systems, Discrete Math., 310 (010), [3] L.Berardi, M.Gionfriddo, R.Rota, Balanced and strongly balanced P k - designs, Discrete Math., 31 (01), [4] L.Berardi, M.Gionfriddo, R.Rota, Perfect octagon quadrangle systems - II, Discrete Math., 31 (01), [5] Y.Chang, G.Lo Faro, A.Tripodi, Tight blocking sets in some maximum packings of λk n, Discrete Math., 308 (008), [6] L.Gionfriddo, M.Gionfriddo, Perfect dodecagon quadrangle systems, Discrete Math., 310 (010), [7] M.Gionfriddo, S.Kucukcifci, L.Milazzo, Balanced and strongly balanced 4-kite systems, Util. Math., (013), [8] M.Gionfriddo, C.C.Lindner, Construction of Steiner quadruples systems having a prescribed number of blocks in common, Discrete Math., 34 (1981), [9] M.Gionfriddo, C.C.Lindner, C.Rodger, -colourings (K 4 e)-designs, Australas. J. Combin., 3 (1991), [10] M.Gionfriddo, G.Lo Faro, -colourings in S(t, t + 1, v), Discrete Math., 111 (1993), [11] M.Gionfriddo, L.Milazzo, A.Rosa, V.Voloshin, Bicolouring Steiner systems S(,4,v), Discrete Math., 83 (004),

9 Equiblocking sets in K, -designs and in K 3,3 -designs 4557 [1] M.Gionfriddo, S.Milici, Z.Tuza, Blocking sets in SQS(v), Comb. Prob. and Comput., 3 (1994), [13] M.Gionfriddo, G.Quattrocchi, Colouring 4-cycle systems with equitably coloured blocks, Discrete Math., 84 (004), [14] M.Gionfriddo, Z.Tuza, On conjectures of Berge and Chvátal, Discrete Math., 14 (1994), [15] G.Lo Faro, A.Tripodi, The Doyen-Wilson theorem for kite systems, Discrete Math., 306 (006), Received: June 3, 013