Connected covering numbers
Jonathan Chappelon, Kolja Knauer, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín
CCONNECTED COVERING NUMBERS
JONATHAN CHAPPELON (cid:63) , KOLJA KNAUER, LUIS PEDRO MONTEJANO,AND JORGE LUIS RAM´IREZ ALFONS´IN
Dedicated to the memory of Michel Las Vergnas
Abstract.
A connected covering is a design system in which the corresponding blockgraph is connected. The minimum size of such coverings are called connected coveringsnumbers . In this paper, we present various formulas and bounds for several parametersettings for these numbers. We also investigate results in connection with
Tur´an systems .Finally, a new general upper bound, improving an earlier result, is given. The latter isused to improve upper bounds on a question concerning oriented matroid due to LasVergnas. Introduction
Let n, k, r be positive integers such that n (cid:62) k (cid:62) r (cid:62)
1. A ( n, k, r ) -covering is a family B of k -subsets of { , . . . , n } , called blocks , such that each r -subset of { , . . . , n } is containedin at least one of the blocks. The number of blocks is the covering’s size . The minimumsize of such a covering is called the covering number and is denoted by C( n, k, r ). Givena ( n, k, r )-covering B , its graph G ( B ) has B as vertices and two vertices are joined ifthey have one r -subset in common. We say that a ( n, k, r )-covering is connected if thegraph G ( B ) is connected. The minimum size of a connected ( n, k, r )-covering is called the connected covering number and is denoted by CC( n, k, r ). Figure 1.
A connected (7 , , Date : January 11, 2015.2010
Mathematics Subject Classification.
Key words and phrases.
Covering design, Tur´an-system, uniform oriented matroid.The second and forth authors were supported by the ANR TEOMATRO grant ANR-10-BLAN 0207.The third author was supported by CONACYT. (cid:63)
Corresponding Author: Phone: +33-467144166. Email: [email protected]. a r X i v : . [ m a t h . C O ] J a n J. CHAPPELON, K. KNAUER, L.P. MONTEJANO, AND J.L. RAM´IREZ ALFONS´IN
The graph corresponding to a connected (7 , , k = r + 1 and thus, we willdenote C( n, r + 1 , r ) (resp. CC( n, r + 1 , r )) by C( n, r ) (resp. by CC( n, r )) for short. Theoriginal motivation to study CC( n, r ) comes from the following question posed by LasVergnas. Question 1.1.
Let U r,n be the rank r uniform matroid on n elements. What is thesmallest number s ( n, r ) of circuits of U r,n , that uniquely determines all orientations of U r,n ? That is, whenever two uniform oriented matroids coincide on these circuits theymust be equal.In [3], Forge and Ram´ırez Alfons´ın introduced the notion of connected coverings andproved that(1) s ( n, r ) (cid:54) CC( n, r ) . The latter was then used to improve the best upper bound, s ( n, r ) (cid:54) (cid:0) n − r (cid:1) , known atthat time due to Hamidoune and Las Vergnas [7]; see also [4] for related results.It turns out that s ( n, r ) is also closely related to C( n, r ). Indeed, by using results in [3, 4]it can be proved that(2) C( n, r ) (cid:54) s ( n, r ) . A proof (needing some oriented matroid notions and thus lying slightly out of scope ofthis paper) of a more general version of the above inequality can be found in [10].Covering designs have been the subject of an enormous amount of research papers (see [6]for many upper bounds and [19] for a survey in the dual setting of Tur´an-systems).Although the construction of block design is often very elusive and the proof of their ex-istence is sometimes tough, here, we will be able to present explicit constructions yieldingexact values and bounds for,C( n, r ) and CC( n, r ) for infinitely many cases. The studyof C( n, r ) and CC( n, r ) seems to be interesting not only for Design Theory but also, inview of Equations (2) and (1), for the implications on the behavior of s ( n, r ) in OrientedMatroid Theory. This relationship was already remarked in [3, Theorem 4.1] where it wasproved that CC( n, r ) (cid:54) n, r ) . The latter can be slightly improved as follows(3) CC( n, r ) (cid:54) n, r ) − , since the graph G associated to a covering with C( n, r ) blocks (and thus with | V ( G ) | =C( n, r )) can be made connected by adding at most C( n, r ) − n, r +1 , r )-connected covering with at most 2C( n, r ) − determine means (up to orientations, bijections, etc.). These (and other) variantsare treated in another paper (see [10]).This paper is organized as follows. In the next section, we recall some basic definitions andresults concerning (connected) coverings and its connection with Tur´an systems neededfor the rest of the paper. In Section 3, we investigate connected covering numbers whenthe value r is either small or close to n . Among other results, we give the exact valuefor CC( n,
2) (Theorem 3.2), for CC( n,
3) for n (cid:54)
12 (Theorem 3.3) and for CC( n, n − n, r ) (Theorem 4.8) ONNECTED COVERING NUMBERS 3 allowing us to improve the best known upper bound for s ( n, r ). We end the paper bydiscussing some asymptotic results in Section 5.2. Basic results
Let n, m, p be positive integers such that n (cid:62) m (cid:62) p . A ( n, m, p ) -Tur´an-system is a family D of p -subsets of { , . . . , n } , called blocks , such that each m -subset of { , . . . , n } containsat least one of the blocks. The number of blocks is the size of the Tur´an-system. Theminimum size of such a covering is called the Tur´an Number and is denoted by T( n, m, p ).Given a ( n, m, p )-Tur´an-system D , with 0 (cid:54) p − m (cid:54) p , its graph G ( D ) has as vertices D and two vertices are joined if they have one 2 p − m -subset in common. We say that a( n, m, p )-Tur´an-system with 0 (cid:54) p − m (cid:54) p is connected if G ( D ) is connected.The minimum size of a connected ( n, m, p )-Tur´an-system is the connected Tur´an Number and is denoted by CT( n, m, p ). By applying set complement to blocks, it can be obtainedthat(4) C( n, k, r ) = T( n, n − r, n − k ) . Moreover, if 0 (cid:54) n − k + r (cid:54) n − k then(5) CC( n, k, r ) = CT( n, n − r, n − k ) . Note that the precondition for (5) is fulfilled if k = r + 1.Most of the papers on coverings consider n large compared with k and r , while for Tur´annumbers it has frequently been considered n large compared with m and p , and oftenfocusing on the quantity lim n →∞ T( n, m, p ) / (cid:0) np (cid:1) for fixed m and p . Thus, for Tur´an-typeproblems, the value C( n, k, r ) has usually been studied in the case when k and r are nottoo far from n .Forge and Ram´ırez Alfons´ın [3] proved that(6) CC( n, r ) (cid:62) (cid:0) nr (cid:1) − r =: CC ∗ ( n, r ) . Moreover, Sidorenko [18] proved that T( n, r + 1 , r ) (cid:62) (cid:0) n − rn − r +1 (cid:1) ( nr ) r . Together with (4), weobtain that(7) CC( n, r ) (cid:62) C( n, r ) = T( n, n − r, n − r − (cid:62) (cid:18) r + 1 r + 2 (cid:19) (cid:0) nr +1 (cid:1) n − r − ∗ ( n, r ) . Combining (6) and (7), together with a straight forward computation we have(8) CC( n, r ) (cid:62) max { CC ∗ ( n, r ) , CC ∗ ( n, r ) } , where the maximum is attained by the second term if and only if r (cid:62) ( n − n, r ) (cid:62) (cid:24) nr + 1 C( n − , r − (cid:25) which can be iterated yielding to(10) C( n, r ) (cid:62) (cid:24) nr + 1 (cid:24) n − r (cid:24) . . . (cid:24) n − r + 12 (cid:25) . . . (cid:25)(cid:25)(cid:25) =: L ( n, r ) . Forge and Ram´ırez Alfons´ın [3, Theorem 4.2] proved that CC( n, r ) (cid:54) (cid:80) n − i = r +1 C( i, r − J. CHAPPELON, K. KNAUER, L.P. MONTEJANO, AND J.L. RAM´IREZ ALFONS´IN (11) CC( n, r ) (cid:54) CC( n − , r ) + C( n − , r − . Results for small and large r In this section, we investigate connected covering numbers for small and large r , that is,when r is very close to either 1 or n . Let us start with the following observations. Remarks . a) CC( n,
0) = 1 since any 1-element set contains the empty set.b) CC( n,
1) = n − K n .c) CC( n, n −
2) = n − n − n, n −
1) = 1 by taking the entire set.All these values coincide with the corresponding covering numbers except in the case r = 1, where C( n,
1) = (cid:100) n (cid:101) .3.1. Results when r is small. For ordinary covering numbers, Fort and Hedlund [5] have shown that C( n,
2) := (cid:100) n (cid:100) n (cid:101)(cid:101) that coincides with the lower bounds given in (10) when the case r = 2.We also have the precise value for the connected case when r = 2. Theorem 3.2.
Let n be a positive integer with n (cid:62) . Then, we have CC( n,
2) = (cid:38) (cid:0) n (cid:1) − (cid:39) . Proof.
Note that the claimed value coincides with the lower bound CC ∗ ( n,
56 87 n − nn −
22 41 3
Figure 2.
Part of the construction proving CC( n, (cid:54) CC ∗ ( n, i − , i, j )for 1 (cid:54) i (cid:54) n and j (cid:62) i + 3. (These are not depicted in the figure.) Now we presentthe gray triangles from left to right. A gray triangle of the form (2 i, i + 1 , i + 4) isconnected to the already presented ones via (2 i − , i, i + 4). Note (as in the figure) thelast triangle may indeed share two edges of already presented triangles, depending on theparity of n . This accounts for the ceiling in the formula. It is easy to check that all edgesare covered. (cid:3) The precise value of C ( n,
3) remains unknown only for finitely many n , see [14, 15, 8].The situation for connected coverings is worse. ONNECTED COVERING NUMBERS 5
Figure 3.
An example proving CC(9 , (cid:54)
28. The circle-vertices are a covering.
Theorem 3.3.
Let n be a positive integer with (cid:54) n (cid:54) . Then, we have CC( n,
3) = (cid:38) (cid:0) n (cid:1) − (cid:39) . Proof.
Note that the claimed value coincides with the lower bound CC ∗ ( n, n (cid:54) , (cid:54)
12 = CC ∗ (7 , ,
2) = 7, and CC(7 ,
3) = 12, we obtain that CC(8 , (cid:54)
19 = CC ∗ (8 , , (cid:54)
28 = CC ∗ (9 , ,
3) = 28and C(9 ,
2) = 12, we conclude that CC(10 , (cid:54)
40 = CC ∗ (10 , , (cid:54)
55 = CC ∗ (11 , ∗ (12 ,
3) we delete the block { , , , } from the covering in Figure 4. One can checkthat this still leaves a covering B , whose graph now has three components. Now, we takethe following (disconnected) (11 , , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } . We add to each of these block the element 12 and thus together with B obtain a (12 , , B (cid:48) . To see that B (cid:48) is connected, note that each of the blocks containing 12 isconnected to a block from B . Moreover, the blocks { , , , } , { , , , } , { , , , } form a triangle and each of them has a neighbor in a different component of G ( B ). Thus, G ( B (cid:48) ) is connected and B (cid:48) has 73 blocks which coincides with CC ∗ (12 , (cid:3) Figure 4.
An example proving CC(11 , (cid:54)
55. The circle-vertices are a covering.
J. CHAPPELON, K. KNAUER, L.P. MONTEJANO, AND J.L. RAM´IREZ ALFONS´IN
Theorem 3.3 supports the following
Conjecture 3.4.
For every positive integer n (cid:62)
4, we haveCC( n,
3) = CC ∗ ( n, . Even, more ambitious,
Question 3.5.
Let n and r be two positive integers such that n (cid:62) r + 1 (cid:62)
4. Is it truethat if CC( n, r ) = CC ∗ ( n, r ) then CC( n (cid:48) , r ) = CC ∗ ( n (cid:48) , r ) for every integer n (cid:48) (cid:62) n ?3.2. Results when r is large.Theorem 3.6. Let n be a positive integer with n (cid:62) . Then, we have CC( n, n −
3) = (cid:18)(cid:6) n (cid:7) (cid:19) + (cid:18)(cid:4) n (cid:5) (cid:19) + 1 . Proof.
The parameter C( n, n −
3) = T( n, ,
2) was determined already by Mantel in1907 [13] and is (cid:0) (cid:100) n (cid:101) (cid:1) + (cid:0) (cid:98) n (cid:99) (cid:1) . Tur´an proved that the unique minimal configuration of setsof size 2 hitting all sets of size 3 of an n -set are the edges of two vertex-disjoint completegraphs K (cid:100) n (cid:101) and K (cid:98) n (cid:99) , see [20].Now, by (4) and (5), the covering corresponding to the Tur´an-system is connected if andonly if the graph whose edges correspond to the blocks of the Tur´an-system is connected.Thus, since the unique optimal construction by Tur´an is not connected but can be madeconnected by adding a single edge connecting the two complete graphs, this is optimalwith respect to connectivity. Therefore, CC( n, n −
3) = T( n, , (cid:3) Proposition 3.7.
Let n (cid:54) = 5 , , , be a positive integer with n (cid:62) . Then, we have CC( n, n − (cid:54) m ( m − m − if n = 3 m,m (2 m − if n = 3 m + 1 ,m (2 m + 1) if n = 3 m + 2 . If n = 5 , , the value of CC( n, n − is one larger than claimed in the formula. Further, CC(8 , ∈ { , } , i.e., it remains open if the above formula has to be increased by oneor not in order to give the precise value.Proof. We will show that a Tur´an-system D verifying the claimed bounds due to Kos-tochka [11] is connected. Indeed the construction of [11] is a parametrized family ofTur´an-systems, each of whose members attains the claimed bound. Our constructionresults from picking special parameters:Assume that n (cid:62)
12 and n is divisible by 3. Split [ n ] into three sets A , A , A of equalsize. Pick special elements x i , y i ∈ A i and denote B i := A i \ { x i , y i } for i = 0 , ,
2. Theblocks of D consist of 3-element sets { a, b, c } of the following forms: L i : a, b, c ∈ A i , T i : a = x i and b, c ∈ A i +1 , T i : a = y i and b, c ∈ B i − ∪ { x i +1 , y i +1 } , T i : a ∈ B i and b, c ∈ B i − ∪ { x i +1 , y i − } where i = 0 , ,
2, and addition of indices is understood modulo 3.Let us now show that D is connected. Clearly, all blocks in a given A i are connected andall 2-element subsets in each A i are covered by a block in this A i . Thus, it suffices to verifythat there are two 2-element sets { e, f } ⊆ A and { e (cid:48) , f (cid:48) } ⊆ A which can be connected bya sequence of blocks of D , because then any block in A containing { e, f } is connected toany block in A containing { e (cid:48) , f (cid:48) } . The connectivity of D then follows by the symmetry ONNECTED COVERING NUMBERS 7 of the construction. Let { e, f } ⊆ B . Take { e, f, y } ∈ T , then { e, y , y } ∈ T , andthen { e, f (cid:48) , y } ∈ T , where f (cid:48) ∈ B , i.e, { y , f (cid:48) } ⊆ B .Now, following [11] deleting any element of such a system yields a Tur´an-system D (cid:48) ofthe claimed size for n (cid:48) = n −
1. We can just delete any x i , since these are not used forconnectivity. Following [11], two elements can be removed from D to obtain a Tur´an-system D (cid:48)(cid:48) of the claimed size for n (cid:48)(cid:48) = n −
2, if the set formed by these two elementsbelongs to exactly n − { x i , x i +1 } , which belongs to exactly n − T i . Again, this preserves connectivity.We are left with the cases n (cid:54)
9. In [18] it is shown that the Tur´an-systems of the claimedsize for n = 9 are exactly the members of the family constructed in [11]. There are exactlytwo such systems:In both cases [9] is split into three sets A , A , A of size 3. In the first system we picka x i ∈ A i and denote A i \ { x i } by B i . The blocks then are the 3-element sets { a, b, c } ofthe following forms: L i : a, b, c ∈ A i , T i : a = x i and b, c ∈ A i +1 , T i : a ∈ B i and b, c ∈ B i − ∪ { x i +1 } .The second system coincides with an instance of a construction due to Tur´an [21]. Itconsists of the following 3-element sets: L i : a, b, c ∈ A i , T i : a ∈ A i and b, c ∈ A i +1 .It is easy to check that both systems are not connected. On the other hand, the secondone can be made connected adding a single block taking one element from each A i . Thisproves the claim for n = 9. Further, removing any vertex not contained in the addedblock, one obtains a connected Tur´an-system for n = 8 with 21 blocks. While there areTur´an-systems showing T(8 , ,
3) = 20 we do not know if there is any such connectedsystem.See Figure 1 for proving our statement for n = 7, Theorem 3.2 for n = 6, and Remark 3.1for n = 4 , (cid:3) A famous conjecture of Tur´an [21] states that the bounds in Proposition 3.7 are bestpossible for C( n, n − n (cid:62)
10 we have(12) C( n, n − (cid:54) CC( n, n − (cid:54) m ( m − m −
1) if n = 3 m,m (2 m −
1) if n = 3 m + 1 ,m (2 m + 1) if n = 3 m + 2 . Tur´an’s conjecture has been verified for all n (cid:54)
13 by [18] and so, by (12), the connectedcovering number can also be determined for these same values.Towards proving Tur´an’s conjecture, it would be of interest to investigate the following.
Question 3.8.
Is it true that one of the inequalities in (12) is actually an equality ?Bounds and precise values for all CC( n, r ) with n (cid:54)
14 are given in Table 1. All the exactvalues previously given in [3] for the same range have been improved by using our aboveresults.Table 1 led us to consider the following.
J. CHAPPELON, K. KNAUER, L.P. MONTEJANO, AND J.L. RAM´IREZ ALFONS´IN r \ n e,t e e e e e e e e e p,t p,u p p p p p [95 l , r ] [121 l , r ]4 1 5 10 t [20 , u ] [32 l , r ] [53 l , r ] [83 l , r ] [124 l , r ] [179 l , r ] [250 l , r ]5 1 6 13 t u [51 l , r ] [96 a , r ] [159 l , r ] [258 l , r ] [401 l , r ]6 1 7 17 t u [84 a , r ] [165 a , r ] [286 l , r ] [501 l , r ]7 1 8 21 t u [126 a , r ] [269 a , r ] [491 l , r ]8 1 9 26 t u [185 a , r ] [419 a , r ]9 1 10 31 t u [259 s , r ]10 1 11 37 t [143 s , u ]11 1 12 43 t
12 1 1313 1
Table 1.
Bounds and values of CC( n, r ) for n (cid:54) r — Upper bound for CC( n, r )(from Equation (11)) e — Exact values for CC( n,
2) (Theorem 3.2) t — Exact values for CC( n, n −
3) (Theorem 3.6) l — Lower bound CC ∗ ( n, r ) p — Some exact values for CC( n,
3) (Theorem 3.3) u — Upper bound for CC( n, n −
4) (Proposition 3.7) s — Lower bound for C( n, r ) (from Equation (9)) a — Lower bounds for C( n, r ) (from [1]) Question 3.9.
Is the sequence (CC( n, i )) (cid:54) i (cid:54) n − unimodal for every n ? or perhaps logarithmically concave ? 4. A general upper bound
Let n and r be positive integers such that n (cid:62) r + 1 (cid:62)
3. Forge and Ram´ırez Alfons´ın [3]obtained the following general upper bound(13) S( n, r ) := (cid:98) n − r +12 (cid:99) (cid:88) i =1 (cid:18) n − ir − (cid:19) + (cid:22) n − r (cid:23) (cid:62) CC( n, r ) . Let us notice that the upper bounds obtained by applying the recursive equation (11), thatwere used in Table 1, are better than the one given by (13). Moreover, by iterating (11)it can be obtained(14) CC( n, r ) (cid:54) n − (cid:88) i = r C( i, r − . Although (14) might be used to get an explicit upper bound for s ( n, r ), it is not clearhow good it would be since that would depend on the known exact values and the upperbounds of C( n, r ) used in the recurrence (and thus intrinsically difficult to compute). On A finite sequence of real numbers { a , a , . . . , a n } is said to be unimodal (resp. logarithmically concave or log-concave ) if there exists a t such that s (cid:54) s (cid:54) · · · (cid:54) s t and s t (cid:62) s t +1 (cid:62) · · · (cid:62) s n (resp. if a i (cid:62) a i − a i +1 holds for every a i with 1 (cid:54) i (cid:54) n − ONNECTED COVERING NUMBERS 9 the contrary, in [3] Equation (13) was used to give the best known (to our knowledge)explicit upper bound for s ( n, r ).In this section, we will construct a connected ( n, r + 1 , r )-covering giving an upper boundfor CC( n, r ) better than S( n, r ) and so, yielding a better upper bound for s ( n, r ) thanthat given in [3]. Theorem 4.1.
Let n and r be positive integers such that n (cid:62) r + 1 (cid:62) . Then CC( n, r ) (cid:54) N( n, r ) , where (15) N( n, r ) := (cid:100) n − r (cid:101) − (cid:88) i =0 ( n − r − i ) (cid:18) r − ir − (cid:19) + (cid:24) n − r (cid:25) − δ C( n − , r − , and δ is the parity function of n − r , that is, δ = (cid:26) if n − r is odd , otherwise . Proof.
From this point on, for any positive integer s , we will denote [ s ] := { , . . . , s } andby (cid:0) [ s ] t (cid:1) the set of all t -subsets of [ s ]. Moreover, for any subset of integers { b , . . . , b s } , wemay suppose that b i < b j for all integers i and j such that 1 (cid:54) i < j (cid:54) s . Case . Suppose that n − r is odd and let m such that n − r = 2 m + 1. We will constructa connected ( r + 2 m + 1 , r + 1 , r )-covering of size m + m (cid:88) i =0 (cid:18) r − ir − (cid:19) (2 m + 1 − i ) . We consider a particular ( r +2 m +1 , r +1 , r )-covering, which is constituted by a large num-ber of blocks but whose associated graph has a small number of connected components.For any i ∈ { , . . . , m } , let N i be the following subset of ( r + 1)-subsets of [ r + 2 m + 1]: N i := { b , . . . , b r +1 } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { b , . . . , b r − } ∈ (cid:0) [ r +2 i − r − (cid:1) b r − = r + 2 i − , b r = r + 2 ib r +1 ∈ { r + 2 i + 1 , . . . , r + 2 m + 1 } . Claim . The set m (cid:91) i =0 N i is a ( r + 2 m + 1 , r + 1 , r ) -covering. Let b = { b , . . . , b r } ∈ (cid:0) [ r +2 m +1] r (cid:1) . If b r − = r − i for some i ∈ { , . . . , m } , then b ⊂ B for some B ∈ N i . The same occurs if b r − = r + 2 i . Claim . The graph G ( N i ) is connected, for any i ∈ { , . . . , m } . Let B = { b , . . . , b r +1 } and C = { c , . . . , c r +1 } in N i . Clearly, B is adjacent to { b , . . . , b r , c r +1 } in G ( N i ). Since b r − = c r − , b r = c r and { d , . . . , d r − , c r − , c r , c r +1 } ∈ N i for all { d , . . . , d r − } ⊂ [ r − i ], then there exists a path from B to C . Claim . There exists a ( r + 1) -subset C i such that G ( N i ∪ C i ∪ N i +1 ) is connected forany i ∈ { , . . . , m − } . Let B i = { , . . . , r − , r − i, r + 2 i, r + 1 + 2 i } ∈ N i and B i +1 = { , . . . , r − , r + 1 +2 i, r + 2 + 2 i, r + 3 + 2 i } ∈ N i +1 . Then, the ( r + 1)-subset C i = { , . . . , r − , r + 2 i, r +1 + 2 i, r + 2 + 2 i } is adjacent to B i and B i +1 in G ( N i ∪ C i ∪ N i +1 ). This concludes theproof of Claim 3.By Claims 4.2, 4.3 and 4.4, we obtain that ( (cid:83) mi =0 N i ) (cid:83) (cid:0)(cid:83) m − i =0 C i (cid:1) is a connected ( r +2 m +1 , r + 1 , r )-covering. Finally, since |N i | = (cid:0) r − ir − (cid:1) (2 m + 1 − i ) for any i ∈ { , . . . , m } ,the theorem holds in this case. Case . Suppose n − r is even and let m be such that n − r = 2 m . We are going toconstruct a ( r + 2 m, r + 1 , r )-connected covering of size m − r − m, r −
2) + m − (cid:88) i =0 (cid:18) r − ir − (cid:19) (2 m − i ) . As already defined in Case 1, we consider the collection N i of ( r + 1)-subsets of [ r + 2 m ]defined by N i := (cid:0) [ r +2 i − r − (cid:1) × { r + 2 i − } × { r + 2 i } × { r + 2 i + 1 , . . . , r + 2 m } for any i ∈ { , . . . , m − } . Let C be a ( r + 2 m − , r − , r − r + 2 m − , r − N m := { B ∪ { r + 2 m − , r + 2 m } | B ∈ C} . Then, one can checkthat m (cid:83) i =0 N i is a ( r + 2 m, r + 1 , r )-covering. Similarly as in the proofs of Claims 4.3 and4.4, it follows that G ( N i ) is connected for any i ∈ { , . . . , m − } and there exists a( r + 1)-subset C i such that G ( N i ∪ C i ∪ N i +1 ) is connected for any i ∈ { , . . . , m − } . Claim . . For any B ∈ N m , there exist i ∈ { , . . . , m − } and C ∈ N i such that B isadjacent to C in the graph G ( N i ∪ N m ) . Let B = { b , . . . , b r − , r +2 m − , r +2 m } ∈ N m . If b r − = r +2 i − i ∈ { , . . . , m − } , then { b , . . . , b r − } ∈ (cid:0) [ r +2 i − r − (cid:1) . Let C = { b , . . . , b r − , r + 2 i − , r + 2 i, r + 2 m } ,by definition C ∈ N i and moreover, since { b , . . . , b r − , r + 2 i − , r + 2 m } ⊂ B and { b , . . . , b r − , r + 2 i − , r + 2 m } ⊂ C , we deduce that B and C are adjacent in thegraph G ( N i ∪ N m ). Either, if b r − = r + 2 i for some i ∈ { , . . . , m − } , we havethat { b , . . . , b r − } ∈ (cid:0) [ r +2 i − r − (cid:1) . We distinguish two cases on the value of b r − . First, if b r − < r + 2 i −
1, then { b , . . . , b r − } ∈ (cid:0) [ r +2 i − r − (cid:1) . Consider now C = { b , . . . , b r − , r + 2 i − , r + 2 i, r + 2 m } . As above, since { b , . . . , b r − , r + 2 i, r + 2 m } ⊂ B and { b , . . . , b r − , r +2 i, r + 2 m } ⊂ C , we deduce that B and C are adjacent in the graph G ( N i ∪ N m ).Finally, suppose that b r − = r + 2 i − α ∈ [ r + 2 i − \ { b , . . . , b r − } and C = { b , . . . , b r − , r + 2 i − , r + 2 i, r + 2 m } ∪ { α } ∈ (cid:0) [ r +2 m ] r +1 (cid:1) . Since { b , . . . , b r − , r + 2 i − , r + 2 i, r + 2 m } ⊂ B and { b , . . . , b r − , r + 2 i − , r + 2 i, r + 2 m } ⊂ C , we deduce that B and C are adjacent in the graph G ( N i ∪ N m ). This concludes the proof of Claim 4.5.Hence, ( (cid:83) mi =0 N i ) ∪ ( (cid:83) m − i =0 C i ) is a connected ( r + 2 m, r + 1 , r )-covering. Since |N i | = (cid:0) r − ir − (cid:1) (2 m − i ) for any i ∈ { , . . . , m − } and |N m | = C( n − , r − (cid:3) Let us illustrate the construction given in the above theorem.
Example 4.6.
N(7 ,
4) = 10. We consider N = { , , } and N = { , , , , , } . It can be checked that N ∪ N is a (7 , , G ( N ) and G ( N ) are connected.Now, by taking C = 12456, it follows that G ( N ∪ C ∪ N ) is connected.We may now show that S( n, r ) > N( n, r ). For this we need first the following Theoremand Proposition. Theorem 4.7.
Let r and n be positive integers such that n (cid:62) r + 1 (cid:62) . Then, S( n, r ) = N( n, r ) + (cid:98) n − r (cid:99) − (cid:88) i =0 (cid:18)(cid:22) n − r (cid:23) − i (cid:19) (cid:18) r − ir − (cid:19) + δ (1 − C( n − , r − , where δ is the parity function of n − r . ONNECTED COVERING NUMBERS 11
Proof.
By induction on n > r . From (13) and (15), the identity is verified for n = r + 1and n = r + 2. Suppose now that the identity is verified for a certain value of n and let D be the difference D := (S( n + 2 , r ) − N( n + 2 , r )) − (S( n, r ) − N( n, r )) . Then S( n + 2 , r ) = N( n + 2 , r ) + (S( n, r ) − N( n, r )) + D. By using (13), we obtainS( n +2 , r ) − S( n, r ) = (cid:98) n − r +12 (cid:99) +1 (cid:88) i =1 (cid:18) n + 2 − ir − (cid:19) + (cid:22) n − r (cid:23) + 1 − (cid:98) n − r +12 (cid:99) (cid:88) i =1 (cid:18) n − ir − (cid:19) − (cid:22) n − r (cid:23) = (cid:18) nr − (cid:19) + 1 . By using (15), we haveN( n +2 , r ) − N( n, r ) = (cid:100) n − r (cid:101) (cid:88) i =0 ( n + 2 − r − i ) (cid:18) r − ir − (cid:19) + (cid:24) n − r (cid:25) + δ C( n, r − − (cid:100) n − r (cid:101) − (cid:88) i =0 ( n − r − i ) (cid:18) r − ir − (cid:19) − (cid:24) n − r (cid:25) + 1 − δ C( n − , r − (cid:100) n − r (cid:101) − (cid:88) i =0 (cid:18) r − ir − (cid:19) + (cid:18) n + 2 − r − (cid:24) n − r (cid:25)(cid:19) (cid:18) r − (cid:6) n − r (cid:7) r − (cid:19) + δ (C( n, r − − C( n − , r − . Moreover, for n − r odd, it follows thatN( n + 2 , r ) − N( n, r ) = (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) + ( δ − (cid:18) n − r − (cid:19) + δ (C( n, r − − C( n − , r − . Therefore D = (cid:18) nr − (cid:19) −(cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) +(1 − δ ) (cid:18) n − r − (cid:19) + δ (C( n − , r − − C( n, r − . From the identity (cid:0) r − ir − (cid:1) = (cid:0) r − ir − (cid:1) − (cid:0) r − ir − (cid:1) , we obtain that (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) = (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) + (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) − (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) = r − (cid:100) n − r (cid:101) (cid:88) i = r − (cid:18) ir − (cid:19) − (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) = (cid:18) r + 2 (cid:6) n − r (cid:7) r − (cid:19) − (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) . Thus, D = (cid:18) nr − (cid:19) − (cid:18) r + 2 (cid:6) n − r (cid:7) r − (cid:19) + (1 − δ ) (cid:18) n − r − (cid:19) + (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) + δ (C( n − , r − − C( n, r − . If n − r is even, then δ = 1 and (cid:18) nr − (cid:19) − (cid:18) r + 2 (cid:6) n − r (cid:7) r − (cid:19) + (1 − δ ) (cid:18) n − r − (cid:19) = (cid:18) nr − (cid:19) − (cid:18) nr − (cid:19) = 0 . Either, if n − r is odd, then δ = 0 and (cid:18) nr − (cid:19) − (cid:18) r + 2 (cid:6) n − r (cid:7) r − (cid:19) + (1 − δ ) (cid:18) n − r − (cid:19) = (cid:18) nr − (cid:19) − (cid:18) n + 1 r − (cid:19) + (cid:18) n − r − (cid:19) = − (cid:18) nr − (cid:19) + (cid:18) n − r − (cid:19) = − (cid:18) n − r − (cid:19) . It follows that D = (cid:100) n − r (cid:101) (cid:88) i =0 (cid:18) r − ir − (cid:19) + ( δ − (cid:18) n − r − (cid:19) + δ (C( n − , r − − C( n, r − (cid:98) n − r (cid:99) (cid:88) i =0 (cid:18) r − ir − (cid:19) + δ (C( n − , r − − C( n, r − . ONNECTED COVERING NUMBERS 13
Now, with the induction hypothesis, we obtainS( n +2 , r ) − N( n +2 , r ) = (S( n, r ) − N( n, r )) + D = (cid:98) n − r (cid:99) − (cid:88) i =0 (cid:18)(cid:22) n − r (cid:23) − i (cid:19) (cid:18) r − ir − (cid:19) + δ (1 − C( n − , r − (cid:98) n − r (cid:99) (cid:88) i =0 (cid:18) r − ir − (cid:19) + δ (C( n − , r − − C( n, r − (cid:98) n − r (cid:99) (cid:88) i =0 (cid:18)(cid:22) n − r (cid:23) + 1 − i (cid:19) (cid:18) r − ir − (cid:19) + δ (1 − C( n, r − . (cid:3) Theorem 4.8.
Let r and n be positive integers such that n − r is an even number. Then, S( n, r ) (cid:62) N( n, r ) + n − r − (cid:88) i =0 (cid:18) n − r − i − (cid:19) (cid:18) r − ir − (cid:19) . Proof.
It is known [6, page 7] that C( n, r ) (cid:54) (cid:0) n − r − (cid:1) +C( n − , r ) . By applying this inequalityrepeatedly we have C( n − , r − (cid:54) n − r − (cid:88) i =0 (cid:18) r − ir − (cid:19) + 1 . Then, we deduce from Theorem 4.7 thatS( n, r ) = N( n, r ) + n − r − (cid:88) i =0 (cid:18) n − r − i (cid:19) (cid:18) r − ir − (cid:19) + 1 − C( n − , r − (cid:62) N( n, r ) + n − r − (cid:88) i =0 (cid:18) n − r − i (cid:19) (cid:18) r − ir − (cid:19) − n − r − (cid:88) i =0 (cid:18) r − ir − (cid:19) = N( n, r ) + n − r − (cid:88) i =0 (cid:18) n − r − i − (cid:19) (cid:18) r − ir − (cid:19) . (cid:3) Asymptotics
In [16] R¨odl uses the probabilistic method to show the existence of asymptotically goodcoverings . Restricted to our case this means thatC( n, r ) (cid:0) nr (cid:1) → r + 1 as n → ∞ . Since CC( n, r ) (cid:54) n, r ) (see [3]) we immediately obtain:CC( n, r ) (cid:0) nr (cid:1) → a (cid:54) r + 1 as n → ∞ . In [3] it was shown that S( n, r ) (cid:0) nr (cid:1) →
12 as n → ∞ and since by Theorem 4.8 the difference N( n, r ) − S( n, r ) is in O ( n r − ) we have the sameasymptotic behavior for N( n, r ).It is however still a topic of research to find explicit constructions witnessing the boundof R¨odl, see [12]. Acknowledgments
Much of this work in particular for the construction of the connected covering designs inFigures 1, 3, 4 strongly benefited from the La Jolla Covering Repository ( ) maintained by Dan Gordon.
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