aa r X i v : . [ m a t h . C O ] J u l On the number of SDRs of a valued ( t, n )-family ∗ Dawei He † and Changhong Lu ‡ Department of Mathematics,East China Normal University,Shanghai 200241, P. R. ChinaJune 30, 2010
Abstract
A system of distinct representatives (SDR) of a family F = ( A , · · · , A n ) is a se-quence ( x , · · · , x n ) of n distinct elements with x i ∈ A i for 1 ≤ i ≤ n . Let N ( F ) denotethe number of SDRs of a family F ; two SDRs are considered distinct if they are differ-ent in at least one component. For a nonnegative integer t , a family F = ( A , · · · , A n )is called a ( t, n )-family if the union of any k ≥ k + t elements. The famous Hall’s Theorem says that N ( F ) ≥ F is a(0 , n )-family. Denote by M ( t, n ) the minimum number of SDRs in a ( t, n )-family. Theproblem of determining M ( t, n ) and those families containing exactly M ( t, n ) SDRswas first raised by Chang [European J. Combin. (1989), 231-234]. He solved the caseswhen 0 ≤ t ≤ t ≥
3. In this paper, we solve the conjecture.In fact, we get a more general result for so-called valued ( t, n )-family.
Keywords.
A system of distinct representatives, Hall’s Theorem, ( t, n )-family.
A system of distinct representatives (SDR) of a family F = ( A , · · · , A n ) is a sequence( x , · · · , x n ) of n distinct elements with x i ∈ A i for 1 ≤ i ≤ n . The famous Hall’s theorem[4] tell us that a family has a SDR if and only if the union of any k ≥ k elements. Several quantative refinements of the Hall’s theorem weregiven in [3, 6, 7]. Their results are all under the assumption of Hall’s condition plus someextra conditions on the cardinalities of A i ’s.Chang [1] extends Hall’s theorem as follows: let t be a nonnegative integer. A family F = ( A , · · · , A n ) is called a ( t, n ) -family if | S i ∈ I A i | ≥ | I | + t holds for any non-emptysubset I ⊆ { , · · · , n } . Denote by N ( F ) the number of SDRs of a family F . Let M ( t, n ) =min { N ( F ) | F is a ( t, n )-family } . Hall’s theorem says that M (0 , n ) ≥
1. In fact, it is easyto know that M (0 , n ) = 1. Chang [1] proved that M (1 , n ) = n + 1 and M (2 , n ) = n + n + 1.He also determined all ( t, n )-families F with N ( F ) = M ( t, n ) for t = 0 , ,
2. Consider the( t, n )-family F ∗ = ( A ∗ , · · · , A ∗ n ), where A ∗ i = { i, n + 1 , · · · , n + t } for 1 ≤ i ≤ n . Then, ∗ Supported in part by National Natural Science Foundation of China (No. 10871166) and ShanghaiLeading Academic Discipline Project (No. B407). † E-mail: [email protected] ‡ E-mail: [email protected] ( F ∗ ) = U ( t, n ) = t X j =0 (cid:18) tj (cid:19)(cid:18) nj (cid:19) j ! . Chang[1] has shown that F ∗ as above is the only (2 , n )-family F with N ( F ) = M ( t, n ),and he conjectured that M ( t, n ) = U ( t, n ) and F ∗ is the only ( t, n )-family F with N ( F ) = M ( t, n ) for all t ≥
3. In 1992, Leung and Wei [5] claimed that they proved the aboveconjecture by means of a comparison theorem for permanents. But Leung and Wei’s proofhas a fatal mistake (see [2]). Hence, the conjecture is still open. In this paper, we solve theconjecture. In fact, we get a more general result for so-called valued ( t, n )-family. In whatfollow, we assume that t ≥ a , · · · , a n ), a family F = ( A , · · · , A n ) is called a valued ( t, n ) -family with valuation ( a , · · · , a n ) if | A i | = a i + t and | S i ∈ I A i | ≥ P i ∈ I a i + t forany | I | ≥
2. Note that a ( t, n )-family F = ( A , · · · , A n ) with N ( F ) = M ( t, n ) must have | A i | = t + 1 for 1 ≤ i ≤ n (see Lemmas 1 and 2 in [1]). Hence, a ( t, n )-family F with N ( F ) = M ( t, n ) is a valued ( t, n )-family with valuation (1 , · · · , F be a valued ( t, n )-family with valuation ( a , · · · , a n ) satisfying | T i ∈ I A i | = t for any | I | ≥
2. Hence, F ∗ is ¯ F withvaluation (1 , · · · , M ′ ( t, n, a , · · · , a n ) = min { N ( F ) | F is a valued ( t, n )-familywith valuation ( a , · · · , a n ) } , and let U ′ ( t, n, a , · · · , a n ) = N ( ¯ F ) = t X j =0 (cid:18) tj (cid:19) j ! X ≤ i < ···
2. The conjecture of Chang [1] is a direct corollary of the conclusion.Some notations are needed. Suppose F is a valued ( t, n )-family with valuation ( a , · · · , a n ).Let N = { , , · · · , n } and B = S i ∈ N A i , and let I x = { i ∈ N | x ∈ A i } and I cx = N − I x for x ∈ B . The degree of x , denoted by deg x , is | I x | . A pair of elements { x, y } ⊆ B is exclusive if I x ∩ I cy = ∅ and I y ∩ I cx = ∅ . An exclusive pair { x, y } is saturated if there exists a subset I ⊆ N satisfying I ∩ I x ∩ I y = ∅ , I ∩ I x ∩ I cy = ∅ , I ∩ I cx ∩ I y = ∅ and | S i ∈ I A i | = P i ∈ I a i + t ;otherwise, we say an exclusive pair { x, y } is unsaturated . { x, y } for a valued ( t, n ) -family Assume that F = ( A , · · · , A n ) is a valued ( t, n )-family with valuation ( a , · · · , a n ) and apair of elements { x, y } is exclusive for F . Let A i ( x, y ) = (cid:26) A i − { x } ∪ { y } , i ∈ I x ∩ I cy ; A i , otherwise.Then we get a new family F xy = ( A ( x, y ) , · · · , A n ( x, y )), but it is possible that F xy is nota valued ( t, n )-family with valuation ( a , · · · , a n ). For any I ⊆ N , by calculating | S i ∈ I A i | and | S i ∈ I A i ( x, y ) | , we can get the relationship between the two values as follows:2 [ i ∈ I A i ( x, y ) | = | S i ∈ I A i | − , I ∩ I x ∩ I y = ∅ , I ∩ I x ∩ I cy = ∅ , I ∩ I cx ∩ I y = ∅ ; | S i ∈ I A i | , otherwise.Hence, F xy is also a valued ( t, n )-family with valuation ( a , · · · , a n ) if and only if { x, y } is unsaturated for F . Furthermore, we have Theorem 1
A valued ( t, n ) -family with valuation ( a , · · · , a n ) satisfying N ( F ) = M ′ ( t, n, a , · · · , a n ) does not contain any unsaturated pair { x, y } . Proof.
Suppose to the contrary that { x, y } is unsaturated for F . Then, F xy is also a valued( t, n )-family with valuation ( a , · · · , a n ). We will prove that N ( F xy ) < N ( F ) and henceleads to a contradiction.Without lose of generality, we can assume that I x ∩ I cy = { , · · · , k } 6 = ∅ , I y ∩ I cx = { k + 1 , · · · , k } 6 = ∅ , I x ∩ I y = { k + 1 , · · · , k } and I cx ∩ I cy = { k + 1 , · · · , n } . So F xy = ( A ( x, y ) , · · · , A n ( x, y )) = ( A − { x } ∪ { y } , · · · , A k − { x } ∪ { y } , A k +1 , · · · , A n ). Let( x , · · · , x n ) be an SDR of F xy . Define a function f from the set of all SDRs of F xy to theset of all SDRs of F as follows:(a) if x i = y for some i ∈ { , · · · , k } and x j = x for some j ∈ { k + 1 , · · · , k } , then( x , · · · , y, · · · , x, · · · , x n ) → ( x , · · · , x, · · · , y, · · · , x n ) . (b) if x i = y for some i ∈ { , · · · , k } and x j = x for all x j , then( x , · · · , y, · · · , x n ) → ( x , · · · , x, · · · , x n ) . (c) otherwise, ( x , · · · , x n ) → ( x , · · · , x n ) .f is clearly one to one. Define F ′ = ( A − { x, y } , · · · , A k − { x, y } , A k +2 − { x, y } , · · · , A n − { x, y } ) . When t ≥ F ′ satisfies the Hall’s condition and has an SDR ( x , · · · , x k , x k +2 , · · · , x n ).Hence, F has an SDR such as( x, x , · · · , x k , y, x k +2 , · · · , x n ) , which is not an f -image of an SDR of F xy , so f is not subjective. Hence, N ( F xy ) < N ( F ). ( t, n ) − family For the set N = { , · · · , n } , we define a relation “ ∼ ′′ on N as follows: i ∼ j if and only ifthere exists a subset I satisfying { i, j } ⊆ I ⊆ N and | S s ∈ I A s | = P s ∈ I a s + t . We claim that“ ∼ ′′ is an equivalent relation on N . It is obvious that “ ∼ ′′ is reflexive and symmetric. If i ∼ j and j ∼ k , then there exist I and J satisfying { i, j } ⊆ I , | S s ∈ I A s | = P s ∈ I a s + t and { j, k } ⊆ J , | S s ∈ J A s | = P s ∈ J a s + t , respectively. Note that I ∩ J = ∅ as j ∈ I ∩ J . Hence, wehave 3 s ∈ I ∪ J a s + t ≤ | [ s ∈ I ∪ J A s | = | ( [ s ∈ I A s ) ∪ ( [ s ∈ J A s ) |≤ | [ s ∈ I A s | + | [ s ∈ J A s | − | [ s ∈ I ∩ J A s |≤ X s ∈ I a s + t + X s ∈ J a s + t − ( X s ∈ I ∩ J a s + t )= X s ∈ I ∪ J a s + t. So we know that | S s ∈ I ∪ J A s | = P s ∈ I ∪ J a s + t and { i, k } ⊆ I ∪ J . It implies that i ∼ k and“ ∼ ′′ is transitive. Hence, “ ∼ ′′ is an equivalent relation. So we can classify N into differentclasses: C , · · · , C m . If an index set I ⊆ N satisfies | S i ∈ I A i | = P i ∈ I a i + t , by the definition of“ ∼ ′′ , we know that I ⊆ C i for some i ∈ { , · · · , m } . Theorem 2
For a valued ( t, n ) -family F with valuation ( a , · · · , a n ) , denote by N SP ( F ) the number of saturated pairs of F , then N SP ( F ) ≤ P ≤ i We use induction on n . When n = 2, the conclusion is obvious.If |B| > n P i =1 a i + t , then by the classification of N under the equivalent relation “ ∼ ′′ ,we get several classes C , · · · , C m and m ≥ 2. Without lose of generality, we can assumethat C = { , · · · , k } , · · · , C m = { k m − + 1 , . . . , n } . We get m subfamilies F , · · · , F m with index sets C , · · · , C m , respectively. According to the preparation before Theorem 2,we know that each saturated pair of F must be saturated for some subfamily F i . Hence, N SP ( F ) ≤ N SP ( F ) + · · · + N SP ( F m ). By induction, N SP ( F ) ≤ X ≤ i 2; (2) | S i ∈ I A i | = P i ∈ I a i + t ; (3) For J ⊂ I , if | J | ≥ 2, then | S i ∈ J A i | > P i ∈ J a i + t . Since |B| = n P i =1 a i + t , the existence of such I holds. Now we use different methodsto discuss two cases I ⊂ N and I = N .For I ⊂ N , without lose of generality, we can assume that I = { k + 1 , · · · , n } , k ≥ B = A , . . . , B k = A k , B k +1 = n S i = k +1 A i , then G = ( B , · · · , B k +1 ) is a valued ( t, k + 1)-family with valuation ( a , · · · , a k , n P i = k +1 a i ). Let { x, y } be an arbitrary saturated pair for F .There are three subcases: (1) { x, y } is saturated for the subfamily ( A , · · · , A k ); (2) { x, y } issaturated for the subfamily ( A k +1 , · · · , A n ); (3) { x, y } is unsaturated for both ( A , · · · , A k )and ( A k +1 , · · · , A n ). It is easy to see that { x, y } in the subcase (1) is also saturated for thefamily G . 4e claim that { x, y } in the subcase (3) is also saturated for G . Since { x, y } is saturatedfor F and unsaturated for both ( A , · · · , A k ) and ( A k +1 , · · · , A n ), there exist ∅ 6 = I ⊆{ , · · · , k } and ∅ 6 = I ⊆ I = { k + 1 , · · · , n } such that | S i ∈ I ∪ I A i | = P i ∈ I ∪ I a i + t and( I ∪ I ) ∩ I x ∩ I y = ∅ , ( I ∪ I ) ∩ I x ∩ I cy = ∅ , ( I ∪ I ) ∩ I y ∩ I cx = ∅ . Since | S i ∈ I ∪ I A i | = P i ∈ I ∪ I a i + t and | S i ∈ I A i | = n P i = k +1 a i + t , using the same discussion in the proof of transitivityof “ ∼ ′′ , we can show that | ( S i ∈ I B i ) ∪ B k +1 | = | ( S i ∈ I A i ) ∪ ( S i ∈ I A i ) | = | ( S i ∈ I ∪ I A i ) ∪ ( S i ∈ I A i ) | = P i ∈ I a i + n P i = k +1 a i + t . Under these circumstances, if { x, y } is not a subset of B k +1 , then { x, y } is saturated for G .Now we will prove that { x, y } is not a subset of B k +1 in two cases: | I | ≥ | I | = 1.If | I | ≥ 2, we claim that I = I . Suppose to the contrary that I ⊂ I . Accordingto I = { k + 1 , · · · , n } , we know that | S i ∈ I A i | = n P i = k +1 a i + t and | S i ∈ I A i | > P i ∈ I a i + t . So | ( S i ∈ I A i ) − ( S i ∈ I A i ) | < n P i = k +1 a i − P i ∈ I a i . Hence, | [ i ∈ I ∪ I A i | = | [ i ∈ I ∪ I A i | + | ( [ i ∈ I − I A i ) − ( [ i ∈ I ∪ I A i ) |≤ | [ i ∈ I ∪ I A i | + | ( [ i ∈ I A i ) − ( [ i ∈ I A i ) | < X i ∈ I ∪ I a i + t + n X i = k +1 a i − X i ∈ I a i = X i ∈ I ∪ I a i + t. It contradicts with the fact that F is a valued ( t, n )-family with valuation ( a , · · · , a n ).Hence, I = I .Now we know that ( I ∪ I ) ∩ I x ∩ I y = ∅ , and hence I ∩ I x ∩ I y = ∅ . Since | S i ∈ I A i | = P i ∈ I a i + t and { x, y } is unsaturated for the subfamily ( A k +1 , · · · , A n ), we have either I ∩ I x ∩ I cy = ∅ or I ∩ I cx ∩ I y = ∅ . Furthermore, we have either I ∩ I x = ∅ or I ∩ I y = ∅ . Therefore, B k +1 = S i ∈ I A i contains at most one of x, y , so { x, y } is not a subset of B k +1 .If | I | = 1, without lose of generality, we can assume that I = { k + 1 } . Since ( I ∪ I ) ∩ I x ∩ I y = ∅ , we know that k + 1 / ∈ I x ∩ I y , which implies that A k +1 contains at most oneof x, y . Assume that y / ∈ A k +1 . Suppose to the contrary that { x, y } is a subset of B k +1 ,then y ∈ S i ∈ I − I A i . By the selection of I and I , we know that y ∈ S i ∈ I ∪ I A i , and hence y / ∈ ( S i ∈ I − I A i ) − ( S i ∈ I ∪ I A i ). Then, | ( [ i ∈ I − I A i ) − ( [ i ∈ I ∪ I A i ) | < | ( [ i ∈ I A i ) − A k +1 | . Since | A k +1 | = a k +1 + t and | S i ∈ I A i | = n P i = k +1 a i + t , we know that5 ( [ i ∈ I A i ) − A k +1 | = | [ i ∈ I A i | − | A k +1 | = n X i = k +2 a i . Therefore, | ( [ i ∈ I A i ) ∪ ( [ i ∈ I A i ) | = | ( [ i ∈ I A i ) ∪ A k +1 ∪ ( [ i ∈ I − I A i ) | = | ( [ i ∈ I A i ) ∪ A k +1 | + | ( [ i ∈ I − I A i ) − ( [ i ∈ I ∪ I A i ) | = X i ∈ I a i + a k +1 + t + | ( [ i ∈ I − I A i ) − ( [ i ∈ I ∪ I A i ) | < X i ∈ I a i + n X i = k +1 a i + t This contradicts with the fact that F is a valued ( t, n )-family with valuation ( a , · · · , a n ).Hence, { x, y } is not a subset of B k +1 .Now we have shown that when I ⊂ N , any saturated pair { x, y } for F is saturated foreither G or the subfamily ( A k +1 , · · · , A n ). Therefore, N SP ( F ) ≤ N SP ( G ) + N SP (( A k +1 , · · · , A n ))by induction, we have N SP ( G ) ≤ X ≤ i N SP ( F ) ≤ P ≤ i 26 ( n P i =1 a i + t )( n P i =1 a i ) − n P i =1 ( a i + t ) a i X ≤ i For a valued ( t, n ) -family F with valuation ( a , · · · , a n ) , denote by N EP ( F ) the number of exclusive pairs of F , then N EP ( F ) ≥ P ≤ i We can assume that n ≥ 2. For an arbitrary element z ∈ B , { x, z } is exclusive for F if and only if x ∈ S i ∈ I cz A i and x / ∈ T i ∈ I z A i . Define D ( z ) = {{ x, z } | { x, z } is exclusive for F } . Therefore, D ( z ) = {{ x, z } | x ∈ [ i ∈ I cz A i − \ i ∈ I z A i } . Let A = { z | deg z = n } and D = {{ x, y } | { x, y } is exclusive for F } . Note that D ( z ) = ∅ if z ∈ A . Then, | D | = 12 X z ∈B | D ( z ) | = 12 X z ∈B−A | D ( z ) | = 12 X z ∈B−A ( | [ i ∈ I cz A i − \ i ∈ I z A i | ) . We first assume that deg z ≥ z ∈ B − A . Then | I z | ≥ | T i ∈ I z A i | ≤ t for all z ∈ B − A . Hence, | D | > X z ∈B−A ( | [ i ∈ I cz A i | − | \ i ∈ I z A i | ) ≥ X z ∈B−A X i ∈ I cz a i . ( ∗ )We point out that the inequality strictly holds as z ∈ T i ∈ I z A i and z / ∈ S i ∈ I cz A i . To calculate P z ∈B−A P i ∈ I cz a i , we construct a weighted bipartite graph G as follows: V ( G ) = V ∪ V , where V = B − A and V = { A , · · · , A n } ; For z ∈ V , if z / ∈ A i , then zA i ∈ E ( G ) and the weightof zA i , denoted by w ( zA i ), is a i . So, X z ∈B−A X i ∈ I cz a i = X z ∈ V X zA i ∈ E ( G ) w ( zA i ) = X A i ∈ V X zA i ∈ E ( G ) w ( zA i ) . ( ∗∗ )Let |A| = a . Obviously, a ≤ t . Each set A i contains a i + t − a elements in B − A andthere are at least n P j =1 a j + t − a elements in B − A . By the construction of G , we know7hat the vertex A i is incident to at least n P j =1 a j − a i edges in G and the weight of each edgeincident to A i is a i . Therefore, X A i ∈ V X zA i ∈ E ( G ) w ( zA i ) ≥ n X i =1 a i ( n X j =1 a j − a i ) = ( n X i =1 a i ) − n X i =1 a i . ( ∗ ∗ ∗ )By above inequalities ( ∗ ), ( ∗∗ ) and ( ∗ ∗ ∗ ), we know that | D | > P ≤ i 3. As the conclusion is obvious when n = 2, we may assume that n ≥ a n = 1, let F = ( A , · · · , A n − ), by induction hypothesis, N EP ( F ) ≥ P ≤ i N EP ( F ) = P ≤ i 2, let F = ( A , · · · , A n − , A n − { x } ), which is a ( t, n )-family with valuation( a , · · · , a n − , a n − N EP ( F ) ≥ P ≤ i 1) and N EP ( F ) = P ≤ i 1) implies that F is ¯ F with valuation( a , · · · , a n − , a n − F are also exclusive for F , | n − S i =1 A i − A n | ≥ n − P i =1 a i , and each element y in n − S i =1 A i − A n is exclusive with x for F and { x, y } isdifferent from any exclusive pair of F . Therefore, N EP ( F ) ≥ X ≤ i 1) + n − X k =1 a k = X ≤ i N EP ( F ) = P ≤ i Theorem 4 M ′ ( t, n, a , · · · , a n ) = U ′ ( t, n, a , · · · , a n ) and ¯ F is the only valued ( t, n ) -family F with valuation ( a , · · · , a n ) satisfying N ( F ) = M ′ ( t, n, a , · · · , a n ) for t ≥ . Applying Theorem 4 to ( t, n )-family, we immediately prove the conjecture of Chang in[1]. References [1] G. J. Chang, On the number of SDR of a ( t, n )-family, Europ. J. Combin. (1989),231-234.[2] G. J. Chang, Corrigendum “A comparison theorem for permanents and a proof of aconjecture on ( t, m )-families”, J. Combin. Theory Ser. A. (1996), 190-192.[3] M. Hall, Jr, Distinct representatives of subsets, Bull. Am. Math. Soc. (1948), 922-926.[4] P. Hall, On reprentatives of subsets, J. London Math. Soc. (1935), 26-30.[5] Joseph Y.-T. Leung and W.-D. Wei, A comparison theorem for permanents and a proofof a conjecture on ( t, m )-families, J. Combin. Theory Ser. A. (1992), 98-112.[6] L. Mirsky, Transversal Theory , Academic Press, New York, 1971.[7] R. Rado, On the number of systems of distinct representatives of sets, J. London Math.Soc.42