aa r X i v : . [ m a t h . C O ] A ug A sharp upper bound for the independencenumber
Peter BorgDepartment of MathematicsUniversity of Malta [email protected]
Abstract An r -graph G is a pair ( V, E ) such that V is a set and E is a familyof r -element subsets of V . The independence number α ( G ) of G is thesize of a largest subset I of V such that no member of E is a subset of I . The transversal number τ ( G ) of G is the size of a smallest subset T of V that intersects each member of E . G is said to be connected iffor every distinct v and w in V there exists a path from v to w (thatis, a sequence e , . . . , e p of members of E such that v ∈ e , w ∈ e p ,and if p ≥ , then for each i ∈ { , . . . , p − } , e i intersects e i +1 ). The degree of a member v of V is the number of members of E that contain v . The maximum of the degrees of the members of V is denoted by ∆( G ) . We show that for any ≤ k < n , if G = ( V, E ) is a connected r -graph, | V | = n , and ∆( G ) = k , then α ( G ) ≤ n − (cid:24) n − k ( r − (cid:25) , τ ( G ) ≥ (cid:24) n − k ( r − (cid:25) , and these bounds are sharp. The two bounds are equivalent. Unless stated otherwise, we shall use small letters such as x to denote non-negative integers or elements (members) of a set. The set { , , . . . } of allpositive integers is denoted by N . For ≤ m ≤ n , we denote { i ∈ N : m ≤ i ≤ n } by [ m, n ] , and if m = 1 , then we also write [ n ] . We take [0] to be the empty set ∅ . For a set X and an integer r ≥ , the set { Y ⊂ X : | Y | = r }
1f all r -element subsets of X is denoted by (cid:0) Xr (cid:1) . Unless stated otherwise,arbitrary sets are assumed to be finite.A pair ( X, Y ) is said to be an r -graph if X is a set and Y is a subsetof (cid:0) Xr (cid:1) . If G is an r -graph ( X, Y ) , then X is represented by V ( G ) and itsmembers are called vertices of G , and Y is represented by E ( G ) and itsmembers are called edges of G . A -graph is also simply called a graph .A subset I of V ( G ) is said to be an independent set of G if no edge of G is a subset of I . The independence number of G , denoted by α ( G ) , is thesize of a largest independent set of G .A subset T of V ( G ) is said to be a transversal of G if T intersects eachedge of G (i.e. e ∩ T = ∅ for each e ∈ E ( G ) ). The transversal number of G ,denoted by τ ( G ) , is the size of a smallest transversal of G .Clearly, the complement V ( G ) \ I of an independent set I of G is a transver-sal of G , and the complement V ( G ) \ T of a transversal T of G is an inde-pendent set of G . By considering the complement of a largest independentset, we obtain τ ( G ) ≤ | V ( G ) | − α ( G ) . By considering the complement of asmallest transversal set, we obtain α ( G ) ≥ | V ( G ) | − τ ( G ) . Thus, we have α ( G ) + τ ( G ) = | V ( G ) | . (1)An r -graph H is said to be a subgraph of G if V ( H ) is a subset of V ( G ) and E ( H ) is a subset of E ( G ) .A vw -path of G is a sequence e , . . . , e p of edges of G such that v ∈ e , w ∈ e p , and if p ≥ , then for each i ∈ [ p − , e i intersects e i +1 . G is said tobe connected if G has a vw -path for every two distinct vertices v and w of G .A component of G is a maximal connected subgraph of G (i.e. a connectedsubgraph of G that is not a subgraph of another connected subgraph). If X , . . . , X c are pairwise disjoint sets whose union is X , then we say that X , . . . , X c partition X . It is easy to see that the following holds. Proposition 1.1 If G , . . . , G c are the distinct components of an r -graph G , then V ( G ) , . . . , V ( G c ) partition V ( G ) , and E ( G ) , . . . , E ( G c ) partition E ( G ) . If v and w are distinct vertices in an edge e of G , then v and w are said tobe adjacent in G , and we say that w is a neighbour of v in G , and vice-versa.An edge e is said to be incident to x if x is a member of e . For v ∈ V ( G ) , N G ( v ) denotes the set of neighbours of v in G , and the degree of v in G ,denoted by d G ( v ) , is the number of edges of G incident to v . The maximumof the degrees of the vertices of G (i.e. max { d G ( v ) : v ∈ V ( G ) } ) is denotedby ∆( G ) . 2n this paper we provide a sharp (i.e. attainable) upper bound for theindependence number of every connected r -graph in terms of the maximumvertex degree. A sharp upper bound for the independence number of every r -graph follows immediately. By (1), this automatically provides a sharplower bound for the transversal number. α ( G ) By (1), an upper bound for the independence number automatically yieldsa lower bound for the transversal number, and vice-versa. More precisely, α ( G ) ≤ | V ( G ) | − a if and only if τ ( G ) ≥ a . Also, α ( G ) ≥ | V ( G ) | − a if andonly if τ ( G ) ≤ a . Various bounds are known for these natural and importantparameters.A classical theorem of Turán [4] says that if G is a -graph and d isthe average degree | V ( G ) | P v ∈ V ( G ) d G ( v ) , then α ( G ) ≥ | V ( G ) | d +1 . Caro [2] andWei [5] independently improved this to α ( G ) ≥ P v ∈ V ( G ) 11+ d G ( v ) . Caro andTuza [3] generalised the Caro-Wei bound to one for every r -graph G , givenby α ( G ) ≥ P v ∈ V ( G ) Q d G ( v ) i =1 (cid:16) − r − i +1 (cid:17) . Alon [1] proved the upper bound τ ( G ) ≤ ln rr | V ( G ) | + | E ( G ) | r for every r -graph G , and he also showed thatthe bound is asymptotically sharp; as explained above, this is equivalent to α ( G ) ≥ (cid:0) − ln rr (cid:1) | V ( G ) | − | E ( G ) | r .We shall instead prove a sharp upper bound for α ( G ) in terms of | V ( G ) | and ∆( G ) .For r ≥ and k ≥ , let f ( r, k ) be the smallest integer n such that thereexists an r -graph G with ∆( G ) = k and | V ( G ) | = n . Proposition 2.1
For every r ≥ and k ≥ , f ( r, k ) = min (cid:26) m ∈ N : (cid:18) mr − (cid:19) ≥ k (cid:27) + 1 . Proof.
Let s = min (cid:8) m ∈ N : (cid:0) mr − (cid:1) ≥ k (cid:9) . Let G be an r -graph such that ∆( G ) = k and | V ( G ) | = f ( r, k ) . Then d G ( v ) = k for some v ∈ G . Let e , . . . , e k be the edges of G that are incident to v . For each i ∈ [ k ] , let e ′ i = e i \{ v } . Let X = S ki =1 e ′ i . So e , . . . , e k ∈ (cid:0) Xr − (cid:1) and hence k ≤ (cid:0) | X | r − (cid:1) . So | X | ≥ s . Since { v } ∪ X ⊆ V ( G ) and v / ∈ X , | V ( G ) | ≥ | X | + 1 ≥ s + 1 . So f ( r, k ) ≥ s + 1 . Now, since k ≤ (cid:0) sr − (cid:1) , we can choose k distinct mem-bers a , . . . , a k of (cid:0) [ s ] r − (cid:1) . For each i ∈ [ k ] , let a ′ i = a i ∪ { s + 1 } . Let H = ([ s +1] , { a ′ i : i ∈ [ k ] } ) . Then H is an r -graph with ∆( H ) = d G ( s +1) = k and | V ( G ) | = s + 1 . So f ( r, k ) ≤ s + 1 . Since f ( r, k ) ≥ s + 1 , we actually3ave f ( r, k ) = s + 1 . ✷ In the next section we construct a connected r -graph U n,r,k with ∆( U n,r,k ) = k , | V ( U n,r,k ) | = n and α ( U n,r,k ) = n − l n − k ( r − m for every r ≥ , k ≥ and n ≥ f ( r, k ) (see Construction 3.5); we take U r,r, = ([ r ] , { [ r ] } ) . The followingis our main result, which is also proved in the next section. Theorem 2.2
Let r ≥ , k ≥ and n ≥ f ( r, k ) . If G is a connected r -graphsuch that | V ( G ) | = n and ∆( G ) = k , then α ( G ) ≤ n − (cid:24) n − k ( r − (cid:25) = (cid:22) ( k − n + 1 k ( r − (cid:23) , and equality holds if G = U n,r,k . A graph that consists of only one vertex is called a singleton . For a graph G , we denote the set of non-singleton components of G by C ( G ) . Corollary 2.3
For every r -graph G , α ( G ) ≤ | V ( G ) | − X H ∈C ( G ) (cid:24) | V ( H ) | − H )( r − (cid:25) , and equality holds if for each H ∈ C ( G ) , H is a copy of U | V ( H ) | ,r, ∆( H ) . Proof.
Let s be the number of singleton components of a graph G . If C ( G ) = ∅ then α ( G ) = s = n . Suppose C ( G ) = ∅ . Clearly, ∆( H ) ≥ foreach H ∈ C ( G ) . By Theorem 2.2, for any H ∈ C ( G ) with ∆( H ) ≥ we have α ( H ) ≤ | V ( H ) | − l | V ( H ) |− H )( r − m . A connected r -graph K with ∆( K ) = 1 canonly consist of r vertices and an edge containing them (i.e. K is a copy of U r,r, ); thus, for any H ∈ C ( G ) with ∆( H ) = 1 we have α ( H ) = r − | V ( H ) | − l | V ( H ) |− H )( r − m . Now, by Proposition 1.1, we clearly have α ( G ) = s + X H ∈C ( G ) α ( H ) ≤ s + X H ∈C ( G ) (cid:18) | V ( H ) | − (cid:24) | V ( H ) | − H )( r − (cid:25)(cid:19) = s + X H ∈C ( G ) | V ( H ) | − X H ∈C ( G ) (cid:24) | V ( H ) | − H )( r − (cid:25) = n − X H ∈C ( G ) (cid:24) | V ( H ) | − H )( r − (cid:25) , H ∈ C ( G ) is a copyof U | V ( H ) | , ∆( H ) . ✷ By (1), we have the following immediate consequence.
Corollary 2.4
For every r -graph G , τ ( G ) ≥ X H ∈C ( G ) (cid:24) | V ( H ) | − H )( r − (cid:25) , and equality holds if for each H ∈ C ( G ) , H is a copy of U | V ( H ) | ,r, ∆( H ) . We start the proof of Theorem 2.2 by making the following observation.
Lemma 3.1 If I is an independent set of an r -graph G , then X v ∈ V ( G ) \ I d G ( v ) ≥ | E ( G ) | . Proof.
For each v ∈ V ( G ) , let A v be the set of those edges of G that areincident to v ; so | A v | = d G ( v ) . Since I is independent, no edge of G has allits vertices in I ; in other words, each edge of G has at least one vertex in V ( G ) \ I . So E ( G ) = S v ∈ V ( G ) \ I A v . We therefore have | E ( G ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ v ∈ V ( G ) \ I A v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X v ∈ V ( G ) \ I | A v | = X v ∈ V ( G ) \ I d G ( v ) as required. ✷ Lemma 3.2 If G is a connected r -graph, e ∈ E ( G ) , and G ′ = ( V ( G ) , E ( G ) \{ e } ) ,then the number of components of G ′ is at most r . Proof.
Let c be the number of components of G ′ . Let G , . . . , G c be thecomponents of G ′ . Suppose e ∩ V ( G j ) = ∅ for some j ∈ [ c ] . Then G j is acomponent of G . Since G is connected, G = G j . So e ∈ E ( G j ) and hence e ∩ V ( G j ) = e = ∅ , a contradiction. So e ∩ V ( G i ) = ∅ for all i ∈ [ c ] .Thus, since V ( G ) , . . . , V ( G c ) partition V ( G ) (by Proposition 1.1), | e | = P ci =1 | e ∩ V ( G i ) | ≥ P ci =1 c and hence r ≥ c . ✷ orollary 3.3 If G is an r -graph, c is the number of components of G , e ∈ E ( G ) , and G ′ = ( V ( G ) , E ( G ) \{ e } ) , then the number of components of G ′ is at most r + c − . Proof.
Let G , . . . , G c be the components of G . By Proposition 1.1, e ∈ E ( G i ) for some i ∈ [ c ] , and e / ∈ E ( G h ) for each h ∈ [ c ] \{ i } . Let G ′ i =( V ( G i ) , E ( G i ) \{ e } ) . Then the components of G ′ are the components of G ′ i and the graphs in the set { G h : h ∈ [ c ] \{ i }} . By Lemma 3.2, G ′ i has at most r components. So G ′ has at most r + c − components. ✷ . Corollary 3.4 If I is an independent set of a connected r -graph G , then X v ∈ V ( G ) \ I d G ( v ) ≥ | V ( G ) | − r − . Proof.
Let G be a connected r -graph, and let n = | V ( G ) | and m = | E ( G ) | .By Corollary 3.3, each time an edge is removed from an r -graph, the numberof components increases by at most r − . Thus, by removing the m edgesof G from G , the number of components obtained is at most m ( r − ;however, the resultant graph is the empty graph ( V ( G ) , ∅ ) , which has n com-ponents (each being a singleton). So n ≤ m ( r − and hence m ≥ n − r − .The result now follows by Lemma 3.1. Proof of Theorem 2.2.
Let G be a connected r -graph with | V ( G ) | = n and ∆( G ) = k . Let I be a largest independent set of G . By Lemma 3.4, n − r − ≤ X v ∈ V ( G ) \ I d G ( v ) ≤ X v ∈ V ( G ) \ I k = | V ( G ) \ I | k = ( n − | I | ) k. So | I | ≤ n − (cid:16) n − k ( r − (cid:17) . Since | I | is an integer and | I | = α ( G ) , we get α ( G ) ≤ n − l n − k ( r − m as required.We now prove that the upper bound is sharp. Consider the followingconstruction. Construction 3.5
We define a graph U n,r,k as follows. Let p = l n − k ( r − m . So n − p − k ( r −
1) + q for some integer q such that ≤ q ≤ k ( r − . Also, p ≥ since n > . Let n ′ = n − p . So n ′ = ( p − k ( r − − q . Let C ∞ ,q,r − be the Cartesian product N × [ k ] × [ r −
1] = { ( i, j, h ) : i ∈ N , j ∈ [ k ] , h ∈ [ r − .For each ( i, j, k ) ∈ C ∞ ,q,r − , let x i,j,h = ( i − k ( r − −
1) + ( j − r −
1) + h. i ∈ [ p ] , let y i = n ′ + i . For each ( i, j ) ∈ [ p ] × [ k ] , let X i,j = { x i,j,h : h ∈ [ r − } and Y i,j = X i,j ∪ { y i } . We have q = s ( r −
1) + t for some s ∈ { } ∪ [ k ] and t ∈ { } ∪ [ r − , where s < k if t > , and s > if t = 0 . Let X ′ p,s +1 = [ n ′ − r + 2 , n ′ ] and Y ′ p,s +1 = X ′ p,s +1 ∪ { y p } . Let M n,r,k = { Y i,j : ( i, j ) ∈ ([ p − × [ k ]) ∪ ( { p }× [ s ]) }∪{ Y ′ p,s +1 } . If n > k ( r − ,then p ≥ and we take U n,r,k to be ([ n ] , M n,r,k ) . Suppose n ≤ k ( r −
1) + 1 .Then p = 1 , y = n , M n,r,k ⊂ { A ∈ (cid:0) [ n ] r (cid:1) : n ∈ A } , | M n,r,k | ≤ s + 1 ≤ k ,and, since n ≥ f ( r, k ) , k ≤ (cid:0) n − r − (cid:1) by Proposition 2.1. Thus, we can choose asubset S n,r,k of { A ∈ (cid:0) [ n ] r (cid:1) : n ∈ A } (note that this is a set of size (cid:0) n − r − (cid:1) ) suchthat M n,r,k ⊆ S n,r,k and | S n,r,k | = k , and we take U n,r,k = ([ n ] , S n,r,k ) .We now conclude the proof. Let U = U n,r,k and M = M n,r,k . We have x , , = 1 < · · · < x , ,r − = r − < x , , = r < · · · < x ,k,r − = k ( r − x , , < · · · < x ,k,r − = 2 k ( r − − x , , < · · · < x ,k,r − = 3 k ( r − − x , , < · · · < x ,k,r − = 4 k ( r − − ...So X , ∪ · · · ∪ X ,k = [ k ( r − , X , ∪ · · · ∪ X ,k = [ k ( r − , k ( r − − , X , ∪· · ·∪ X ,k = [2 k ( r − − , k ( r − − , and so on. Let T = { y i : i ∈ [ p ] } and J = ( S Y ∈ M Y ) \ T . Then J = (cid:16)S ( i,j ) ∈ [ p − × [ k ] X i,j (cid:17) ∪ (cid:16)S j ∈ [ s ] X p,j (cid:17) ∪ X ′ p,s +1 = [ n ′ ] and T = [ n ′ + 1 , n ] . Thus, each member of [ n ] is in some edgeof U . Now T is a transversal of U . If k ≤ k ( r −
1) + 1 , then T = { y } .If k > k ( r −
1) + 1 , then Y ,k , Y , , . . . , Y p − , , Y p − ,k , Y p, is a y y p path of U such that each member of T is a member of some edge in the path (note thatfor each i ∈ [ p − , Y i,k ∩ Y i +1 , = { x i,k,r − } and Y i, ∩ Y i,k = { y i } ). Thus,since each edge of U is incident to some member of T , U is connected. Byconstruction, U is an r -graph, | V ( U ) | = n , and ∆( U ) = d G ( y ) = k (notethat Y ′ p,s +1 = Y p,s if t = 0 , and recall that y = n if n ≤ k ( r −
1) + 1 ).So α ( U ) ≤ n − l n − k ( r − m = n ′ = | J | . By (1), J is an independent set of U since J = [ n ] \ T . So α ( U ) ≥ | J | . Since α ( U ) ≤ | J | , we actually have α ( U ) = | J | = n − l n − k ( r − m . ✷ References [1] N. Alon, Transversal numbers of uniform hypergraphs, Graphs and Com-bin. 6 (1990) 1–4. 72] Y. Caro, New results on the independence number, Technical Report,Tel-Aviv University, 1979.[3] Y. Caro, Z. Tuza, Improved lower bounds on kk