New Lower Bounds for the Number of Blocks in Balanced Incomplete Block Designs

Transcription

1 New Lower Bounds for the Number of Blocs in Balanced Incomplete Bloc Designs MUHAMMAD A KHAN ABSTRACT: Bose [1] proved the inequality b v+ r 1 for resolvable balanced incomplete bloc designs (RBIBDs) and Kageyama [4] improved it for RBIBDs which are not affine resolvable In this note we prove a new lower bound on the number of blocs b that holds for all BIBDs We further prove that for a significantly large number of BIBDs our bound is tighter than the bounds given by the inequalities of Bose and Kageyama 1 INTRODUCTION Let D be a balanced incomplete bloc design (BIBD) with parameters (v, b, r,, λ) The following results follow immediately from definition b = vλ, r( 1) = λ(v 1) A less obvious relation is Fisher s inequality [3] b v (or equivalently r ) In his classical paper on combinatorial designs Bose [1] strengthened Fisher s inequality for RBIBDs proving that b v + r 1 Later Michael [6] proved the same inequality under the weaer assumption v = n, for some positive integer n Stanton [7] showed that a necessary and sufficient condition for Bose s inequality is r λ + (which is weaer than v = n) Over the years Bose s inequality has been improved under stronger assumptions For instance Kageyama [4] showed that b v+ r 1 for resolvable designs which are not affine resolvable However, so far the lower bound given by Bose s inequality has not been improved generally The question that motivated this paper is the following: Can we establish a lower bound for b that wors for all BIBDs and at the same time improves Bose s bound for a significant number of BIBDs? Such results are important as they can help in answering questions regarding the existence of BIBDs Here we prove the following theorems: 000 Mathematics Subject Classification: 05B05 Key Words and Phrases: Balanced incomplete bloc designs, Bose s inequality, Fisher s inequality, 1-designs, leave of a t-design, complement of a BIBD 1

2 Theorem 11 For a BIBD with parameters (v, b, r,, λ) ( v ) 3 a) if D is nontrivial then b + r λ v ( v ) b) if D is nontrivial and r v 1 then b + r λ v 1 We prove theorem 11 in section, establishing some new inequalities for BIBD s along the way In section 3 we compare the lower bounds given in theorem 11 with Bose s lower bound for fifty admissible parameter sets listed in the Handboo of Combinatorial Designs [] We also demonstrate that even in the special case of resolvable designs which are not affine resolvable our bounds are tighter than Kageyama s bound PROOF OF THEOREM 11 AND RELATED RESULTS A BIBD with = 1 or = v is trivial So 1< < v for any nontrivial BIBD We begin with some preliminary but significant results Lemma 1 Let D be a BIBD with parameters (v, b, r,, λ) Then a) b λv m m 3 3 b) if D is nontrivial then b < λv for m > 3 In particular b < λv m m 1 c) if D is nontrivial then < λv for m > 3 In particular 3 < λv Proof vr λ( v 1) a) b λv = λv = v λv 1 λv λv λv+ λv λv λv = = b) It suffices to prove the result for m = 3 Observe that the function strictly increasing for x So for v < 1 v 1 λvv ( 1) ( 1) b < λv 3 3 < λv 3 3 But D is nontrivial so we always have c) Follows from b) and Fisher s inequality A similar argument leads to the corresponding result for 1-designs x f( x) = is x 1

3 Lemma If D is a 1-(V, B, R, K, Λ) design then m m a) for m > we have BK Λ V In particular BK Λ V m m 1 b) for m > and B V we have K Λ V In particular K Proof Follows from BK = RV =ΛV and V K Λ V Note that for lemma b) we explicitly need to assume b v as this does not hold in general for 1-designs The question may be ased as to whether λv holds for a BIBD with parameters 3 (v, b, r,, λ) Clearly the bound in theorem 11 can be improved if we replace λv with λv In fact a direct proof of λv together with lemma 1 b) would prove Fisher s inequality However neither λv nor λv is obvious from lemma 1 Here we shall use a variant of lemma to establish the relation λv for a large class of BIBDs The proof uses the idea of leave of a t-design Let E be a t-(v,, λ) design L with b blocs and replication number r The leave E of E is a (t 1)-(v 1,, λ) design with (b r) blocs (see [, p 10] for complete definition) Theorem 3 If D is a BIBD with parameters (v, b, r,, λ) and b v+ r 1 then λ( v 1) L Proof The leave D of D is a 1-design Since b r v 1 therefore we can apply lemma L b) to D so that λ( v 1) The above proof provides an instance where 1-designs have been used to prove a theorem about BIBDs (-designs) This is extremely rare in literature as BIBDs have a rich structure compared to 1-designs Now we can return to our main results For the proof of theorem 11 recall that the complementary design of a BIBD with parameters (v, b, r,, λ) is a BIBD with parameters (v, b, b r, v, b r+ λ ) provided b r+ λ 0 Proof of Theorem 11 a) Let D be a BIBD satisfying the hypothesis As D is nontrivial therefore v and λvv ( 1) λ( v 1) ( v 1) v b r+ λ = + λ λ = 1 0 ( 1) ( 1) + ( 1) So C D exists and has parameters (v, b, b r, v, m = 3 of lemma 1 c) gives result follows since b is a positive integer ( v ) ( b r ) v 3 < + λ and so b) Let (v *, b *, r *, *, λ * ) = (v, b, b r, v, b r+ λ ) b r+ λ ) Now applying the case ( v ) 3 b> + r λ The v 3

4 If r v 1 then b (b r) v 1 giving b * r * v * 1 Thus theorem 3 is applicable and we have v + or b + r λ v 1 ( v ) ( b r λ)( v 1) ( ) 3 SHARPNESS OF LOWER BOUNDS The following table lists Bose s lower bound and the bounds given by theorem 11 for admissible parameter sets listed in the [, pp 36-58] ordered lexicographically by r, and λ The first column gives the serial number assigned in The Handboo of Combinatorial Designs The entries in bold indicate that the bound(s) given by theorem 11 are more stringent than the one given by Bose s inequality The dash specifies that the bound is not applicable No Admissible Parameter Sets TABLE 1 Comparison of Lower Bounds Part (a) Theorem 11 Part (b) Bose v + r 1 1 (7,7,3,3,1) (9,1,4,3,1) (13,13,4,4,1) (6,10,5,3,) (16,0,5,4,1) (8,8,14,4,6) (15,4,14,5,4) (36,84,14,6,) (15,35,14,6,5) (85,170,14,7,1) (8,7,18,7,4) (64,144,18,8,) (145,90,18,9,1) (73,146,18,9,) (49,98,18,9,3) (85,105,1,17,4) (10,140,1,18,3)

5 No Admissible Parameter Theorem 11 Bose Sets Part (a) Part (b) v + r (190,10,1,19,) (400,40,1,0,1) (41,41,1,1,1) (59,55,4,3,1) (553,553,4,4,1) (77,77,4,4,) (185,185,4,4,3) (139,139,4,4,4) (55,99,7,15,7) (460,690,7,18,1) (153,31,7,18,3) (5,78,7,18,9) (91,117,7,1,6) (61,366,30,5,) (41,46,30,5,3) (31,186,30,5,4) (5,150,30,5,5) (1,16,30,5,6) (17,68,3,8,14) (145,464,3,10,) (5,80,3,10,1) (33,96,3,11,10) (177,47,3,1,) (69,138,34,17,16) (35,70,34,17,16) (715,1105,34,,1) (69,10,34,3,11)

6 No Admissible Parameter Theorem 11 Bose Sets Part (a) Part (b) v + r (154,187,34,8,6) (13,78,36,6,15) (17,1116,36,7,1) (8,144,36,7,8) (64,88,36,8,4) (,99,36,8,1) Looing at table 1 it seems that very few BIBDs satisfy the condition r v 1 But the following theorem shows that this is not the case Indeed if a nontrivial BIBD satisfies C Bose s inequality then either D or D must satisfy this condition Theorem 31 If D is a nontrivial BIBD with parameters (v, b, r,, λ) such that C b v+ r 1 and r < v 1 Suppose the complementary design D has parameters (v *, b *, r *, *, λ * ) then r * v * 1 Proof Here (v *, b *, r *, *, λ * ) = (v, b, b r, v, b r+ λ ) Now b r v 1 so that r * v * 1 It is worth noting that the lower bound in part a) of theorem 11 is satisfied by all nontrivial BIBDs regardless of their parameters This is quite unique as only Fisher s inequality is nown to offer such a general meaningful lower bound At the same time both of our inequalities improve Bose s inequality for several BIBDs In particular as r, and λ increase lexicographically our bounds become considerably better than Bose Interestingly even Kageyama s bound is not always stronger than any of our lower bounds For instance consider the RBIBD with parameters (16, 140, 35, 4, 7) which is not affine resolvable [4, Example (iii)] The inequalities in theorem 11 a) and b) respectively say b 71 and b 73 whereas Kageyama s inequality only says b 65 Concluding Remars: In this paper we have established two new lower bounds for the number of blocs in a balanced incomplete bloc design which improve all nown lower bounds for a large number of BIBDs At the same time we have suggested a possible approach for proving Fisher s inequality combinatorially without using linear algebra Currently we are extending the methods described here to the more general setting of t- designs [5] There is plenty of room for improvement even for the lower bounds proved for BIBDs We conjecture the following: 6

7 Open Problem 31 If D is a nontrivial BIBD with parameters (v, b, r,, λ) and ( v ) b v+ r 1 then b + r λ v 1 So far the computational evidence supports this conjecture Acnowledgements: The author is thanful to King Fahd University of Petroleum & Minerals for continuous research support REFERENCES [1] Bose, RC, A note on the resolvability of Balanced Incomplete Bloc Designs, Sanhya 6 (194) [] Colbourn, CJ, Dinitz, JH, Handboo of Combinatorial Designs, nd Ed (007) Chapman and Hall/CRC [3] Fisher, RA, An examination of the different possible solutions of a problem in incomplete blocs, Ann Eugenics 10 (1940) 5-75 [4] Kageyama, S, An improved inequality for balanced incomplete bloc designs, Ann Math Statist 4 (1971) [5] Khan, MA, Improved bounds on the number of blocs in t-designs, submitted [6] Michail, WF, An inequality for balanced incomplete bloc design, Ann Math Statist 31 (1960) 50-5 [7] Stanton, RG, A note on BIBDS, Ann Math Statist 8 (1957) [8] Wallis, WD, Combinatorial Designs, (1988) Mercel Deer, Inc PREPARATORY YEAR MATHEMATICS PROGRAM KING FAHD UNIVERSITY OF PETROLEUM & MINERALS DHAHRAN 3161, SAUDI ARABIA malihan@fupmedusa 7