Sum-distinguishing number of sparse hypergraphs
aa r X i v : . [ m a t h . C O ] F e b Sum-distinguishing number of sparse hypergraphs
Maria Axenovich ∗ , Yair Caro † , and Raphael Yuster ‡ Abstract
A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and thesums of labels taken over the distinct hyperedges are distinct. Let s ( H ) be the smallest integer N such that there is a sum-distinguishing labeling of H with each label at most N . The largestvalue of s ( H ) over all hypergraphs on n vertices and m hyperedges is denoted s ( n, m ). Weprove that s ( n, m ) is almost-quadratic in m as long as m is not too large. More precisely, thefollowing holds: If n ≤ m ≤ n O (1) then s ( n, m ) = m w ( m ) , where w ( m ) is a function that goes to infinity and is smaller than any polynomial in m .The parameter s ( n, m ) has close connections to several other graph and hypergraph func-tions, such as the irregularity strength of hypergraphs. Our result has several applications,notably: • We answer a question of Gy´arf´as et al. whether there are n -vertex hypergraphs withirregularity strength greater than 2 n . In fact we show that there are n -vertex hypergraphswith irregularity strength at least n − o (1) . • In addition, our results imply that s ∗ ( n ) = n /w ( n ) where s ∗ ( n ) is the distinguishingclosed-neighborhood number, i.e., the smallest integer N such that any n -vertex graphallows for a vertex labeling with positive integers at most N so that the sums of labels ondistinct closed neighborhoods of vertices are distinct. For a hypergraph H = ( V, E ), we say that a labeling f : V → N is sum-distinguishing or simply distinguishing if s ( e ) = s ( e ′ ) for any two distinct hyperedges e, e ′ ∈ E , where s ( e ) = P v ∈ e f ( v ).Let s ( H ) be the smallest integer N such that there is a distinguishing labeling of H with each labelat most N . Note that s ( H ) is well-defined by assigning vertex labels equal to distinct powers of2. Distinguishing labelings can be viewed as number-theoretic constructions extending Sidon setsto non-complete, non-uniform hypergraphs. Using common notation, a B h [1]-Sidon set is a set X of integers such that for any integer q , there is at most one subset X ′ of X , | X ′ | = h , so that the ∗ Department of Mathematics, Karlsruhe Institute of Technology, Germany † Department of Mathematics, University of Haifa-Oranim, Israel ‡ Department of Mathematics, University of Haifa, Israel X ′ is q . So, a B h [1]-Sidon set corresponds to a distinguishing labeling of acomplete h -uniform hypergraph. On the other hand, distinguishing labelings of hypergraphs areclosely connected to several “distinguishing” type parameters of graphs and hypergraphs that wediscuss in more detail later. Let s ( n, m ) = max { s ( H ) : | V ( H ) | = n, | E ( H ) | = m } . Namely, s ( n, m ) is the largest value of s ( H ) over all hypergraphs on n vertices and m hyperedges.Observe first that for the largest possible value of m , namely m = 2 n − s ( n, n −
1) = 2 n − . So,in particular, we have that s ( n, m ) is linear in the number of edges whenever m = Θ(2 n ). On theother hand, for general m , a standard probabilistic argument shows that s ( n, m ) = O ( m ). So, itseems of interest to study the dependence of s ( n, m ) on m whenever the hypergraph is relativelysparse. Our main result does just that. We prove, perhaps surprisingly, that for hypergraphs withpolynomially many edges, s ( n, m ) is neither linear nor quadratic. In fact, we prove that in thisregime, s ( n, m ) is almost-quadratic in m . Theorem 1. If n ≤ m ≤ n O (1) then s ( n, m ) = m w ( m ) , where w ( m ) is a function that goes to infinity and is smaller than any polynomial in m . Moreformally, for any C > , ǫ > , there is n such that for any n > n , and any m satisfying n ≤ m ≤ n /ǫ , we have that m − ǫ ≤ s ( n, m ) ≤ m /C . The upper bound in the proof of Theorem 1 relies on several probabilistic arguments, some ofwhich are rather delicate. For the lower bound, we extend an approach of Bollob´as and Pikhurko[8] used for 2-uniform hypergraphs (i.e. graphs) and their distinguishing labelings.Our main result has several applications that we next describe. Our first application is to theproblem of distinguishing the vertices of a graph by sums of labels on closed neighborhoods. Fora graph G = ( V, E ), and a vertex v ∈ V ( G ), the open neighborhood of v is N ( v ) = { u ∈ V ( G ) : uv ∈ E ( G ) } ; the closed neighborhood of v is N [ v ] = { v } ∪ N ( v ). For a vertex labeling f of G and v ∈ V ( G ), let s ∗ ( v ) = s ∗ f ( v ) = P x ∈ N [ v ] f ( x ). The labeling f is called vertex sum-distinguishing ifit uses positive integers and s ∗ ( v ) = s ∗ ( u ) for any u, v ∈ V ( G ) such that N [ u ] = N [ v ]. Let s ∗ ( G )be the smallest integer k such that there is a vertex sum-distinguishing labeling of G with a largestlabel k and let s ∗ ( n ) be the maximum of s ∗ ( G ) taken over all graphs with n vertices.Let s ( n ) = s ( n, n ). We observe that the parameters s ( n ) and s ∗ ( n ) are closely connected.Indeed, for a graph G = ( V, E ) consider a hypergraph H = H G on a vertex set V with hyperedgescorresponding to the closed neighbourhoods of vertices of G . We see that s ∗ ( G ) = s ( H ). Note thatthe number of hyperedges in H G is at most n . The following result is an immediate consequenceof Theorem 1 and Lemma 1, in which we prove that s ( n/ ≤ s ∗ ( n ) ≤ s ( n ). Thus, we obtain:2 orollary 1. We have that s ∗ ( n ) = n w ( n ) , where w ( n ) is a function that goes to infinity and is smaller than any polynomial in n . Moreformally, for any C > , ǫ > , there is n such that for any n > n , n − ǫ ≤ s ∗ ( n ) ≤ n /C . The proof of Theorem 1 (and hence Corollary 1) yields an efficient randomized algorithm forfinding a corresponding labeling. For graphs with given maximum and minimum degrees, weprovide a more specific result which also yields an efficient deterministic algorithm.
Theorem 2.
Let G be a nonempty n -vertex graph with maximum degree ∆ , minimum degree δ ,and the largest number of vertices with pairwise distinct closed neighborhoods equal to n ′ . Let d ( v ) denote the degree of v . Then n ′ + δ ∆ + 1 ≤ s ∗ ( G ) ≤ max { ( n − d ( v ) − d ( v ) + 1) + 2 : v ∈ V ( G ) } ≤ (∆ + 1) n . Observe that the upper bound of Theorem 2 is weaker than the upper bound in Corollary 1whenever ∆ = Θ( n ). In fact, it only gives s ∗ ( n ) ≤ n / s ∗ , weconsider the case where the graph is a tree. For a tree T and its vertex u , let L ( u ) be the set ofleaves adjacent to u . Let L ( T ) = max {| L ( u ) | : u ∈ V ( T ) } . Theorem 3.
Let T be a tree with n ≥ vertices. Then s ∗ ( T ) ≤ n − − L ( T ) , moreover thisbound is tight for stars. The parameter s ( n ) is closely related to the notion of irregularity strength of hypergraphs. Thereis extensive literature on irregularity strengths of graphs, a notion first introduced (for graphs) byChartrand et al. [9], see also for example Nierhoff [13] and Blokhuis and Sz˝onyi [7], Balister et al.[4], as well as the survey by Gallian [10]. To define irregularity strength, consider an edge-labeling f of a hypergraph H with positive integers and for each vertex x compute s ( x ), the sum of labelsover all hyperedges containing x . The labeling is irregular , if the sums s are distinct for all vertices.The smallest value of a largest label used in an irregular labeling of H is denoted irr ( H ) and irr ( n ) is the largest value of irr ( H ) over all n -vertex hypergraphs. Note that irr ( H ) correspondsto s ( H ∗ ), where H ∗ is the dual hypergraph of H . Recall that for a hypergraph H = ( V, E ), thedual hypergraph H ∗ has vertex set E and edge set {{ e ∈ E : e ∋ x } : x ∈ V } . Gy´arf´as et al.[11] provided upper bounds on irr ( H ) and stated that “it is not known whether irr ( n ) ≥ n .” Aconsequence of our result gives a better lower bound irr ( n ) ≥ n − ǫ for any positive ǫ and sufficientlylarge n , and in particular, answers their question. Theorem 4.
For any ǫ > , there is n such that for any n > n , irr ( n ) ≥ n − ǫ . We mention a few other closely related problems that have been studied. There is yet anotherparameter, similar to s ( H ), introduced by Bhattacharya et al. [6] and called a discriminator wherethe goal is to assign non-negative integer labels to vertices of a hypergraph such that the sumson the hyperedges are distinct, and positive. While our original motivation was to distinguish3he vertices of a graph via sums on closed neighborhoods, there is a similar problem restrictedto pairs of vertices that are adjacent, i.e., so-called adjacent vertex sum-distinguishing number,that was studied for closed neighborhoods by Axenovich et al. [3] and for open neighborhoods byBartnicki et al. [5], who use also an unpublished observation by Norin. These above-mentionedadjacency-dependent parameters can however be upper-bounded by a function of the maximumdegree, independent of the number of vertices of the graph. Finally, we mention that distinguishinglabelings of graphs were also studied by Ahmad et al. [1].The rest of the paper is structured as follows. In the next section we prove several lemmasthat are required for our theorems. In particular, Lemma 1 comparing s ∗ ( n ) and s ( n ), Lemma 2,which is the main ingredient in the lower bound on s ( n, m ) as it implies the existence of a certain(randomly constructed) hypergraph H with large s ( H ), and Lemma 3 about the distribution ofthe sum of discrete random variables, that we use for the upper bound on s ( n, m ). In Section 3we prove Theorem 1, our main result. Section 4 contains the proofs of Theorems 2, 3, and 4. Thefinal section consists of concluding remarks and open problems. This section consists of several lemmas facilitating the proof of our main theorems. For a positiveinteger x , we use the notation [ x ] = { , . . . , x } . Our first lemma relates s ∗ ( n ), s ( n ), and s ∗ (2 n ). Lemma 1.
For any n ≥ , we have s ∗ ( n ) ≤ s ( n ) ≤ s ∗ (2 n ) .Proof. Let G be an n -vertex graph with s ∗ ( G ) = s ∗ ( n ). As mentioned in the introduction, con-sider a hypergraph H on the vertex set V = V ( G ) with hypergedges corresponding to the closedneighborhoods of vertices in G . Since a labeling f of V is vertex sum-distinguishing in G if andonly if it distinguishing in H , we have that s ∗ ( n ) = s ∗ ( G ) = s ( H ) ≤ s ( n ) . On the other hand,consider a hypergraph H on a vertex set B = { b , . . . , b n } and with n hyperedges e , . . . , e n , suchthat s ( H ) = s ( n ). Let G be a graph on vertex set A ∪ B , where A = { a , . . . , a n } , A ∩ B = ∅ ,where A induces a clique with n vertices, B induces an independent set, and a i b j ∈ E ( G ) if andonly if b j ∈ e i . Then we see that if a labeling f is vertex sum-distinguishing in G then, restrictedto B , it is distinguishing in H . Consider such an optimal f , i.e. with a largest label s ∗ ( G ). Since G has 2 n vertices, s ∗ ( G ) ≤ s ∗ (2 n ). Thus, s ( n ) = s ( H ) ≤ s ∗ ( G ) ≤ s ∗ (2 n ). Lemma 2.
For any fixed r ≥ , there is a constant c = c ( r ) such that for every positive integer N it holds that there exists an r -uniform hypergraph H on N vertices such that | E ( H ) | = Θ( N ( r +1) / p log N ) , a nds ( H ) ≥ cN r . Proof.
We are going to extend a result of Bollob´as and Pikhurko [8] on distinguishing labelings ofgraphs to r -uniform hypergraphs. Also note that the inequalities in the lemma’s statement allow Observe that H might have less than n edges since not all closed neighborhoods of G are necessarily distinct,but since adding edges to a hypergraph cannot decrease s , we indeed have s ( H ) ≤ s ( n )
4s to assume, whenever necessary, that N is sufficiently large as a function of r .Proof idea: We provide a lower bound on s ( H ) for a random r -uniform hypergraph H ∼ G r ( N, p ), i.e., a hypergraph on a vertex set [ N ], such that hypergedges are chosen independentlywith probability p . In order to show that s ( H ) > s for a chosen s , we shall consider a fixed labeling f of [ N ] and denote by p ′ the probability that f is distinguishing for H . Now, if it holds that p ′ = o ( s − N ) then we have Pr[ s ( H ) ≤ s ] ≤ s N p ′ = o (1). So, in this case we see that almost surely s ( H ) > s .Let q = √ r · r !, p = q √ ln N / √ N r − , s = ⌊ N r / (2 r · r !) ⌋ and H ∼ G r ( N, p ). Consider alabeling f : [ N ] → [ s ]. For any e ∈ (cid:0) [ N ] r (cid:1) , let s ( e ) = P i ∈ e f ( i ). We estimate p ′ , the probability that f is distinguishing for H .Let H k be the r -uniform hypergraph on a vertex set [ N ], with E ( H k ) = { e ∈ (cid:0) [ N ] r (cid:1) : s ( e ) = k } and denote h k = | E ( H k ) | . Note that for any r -subset of the vertices e ∈ (cid:0) [ N ] r (cid:1) , r ≤ s ( e ) ≤ sr andthat the H k ’s form an edge-decomposition of the complete r -uniform hypergraph on the vertex set[ N ]. Note that f is distinguishing for H if and only if H has at most one edge in each of the H k ’s.We need to consider only those H k ’s that have at least two edges so let K = { k : h k ≥ } . We have p ′ = Pr[ f is distinguishing for H ]= Y k ∈ K Pr[ | E ( H ) ∩ E ( H k ) | ≤ Y k ∈ K (cid:16) (1 − p ) h k + h k p (1 − p ) h k − (cid:17) . We need the following statement that is a routine calculation. If t ≤ t − (cid:0) (1 − p ) t + t p (1 − p ) t − (cid:1) (cid:0) (1 − p ) t + t p (1 − p ) t − (cid:1) ≤ (cid:16) (1 − p ) t +1 + ( t + 1) p (1 − p ) ( t +1) − (cid:17) (cid:16) (1 − p ) t − + ( t − p (1 − p ) ( t − − (cid:17) . (1)Using (1) we can upper-bound the expression for p ′ by the one in which each h k takes an integervalue x or x + 1, for some x . Let there be b of x ’s and | K | − b of ( x + 1)’s, so bx + ( | K | − b )( x + 1) = P k ∈ K h k = h . Assume that xb ≥ h/ | K | − b )( x + 1) ≥ h/ p ′ = Y k ∈ K (1 − p ) h k + h k p (1 − p ) h k − ≤ (1) (cid:0) (1 − p ) x + xp (1 − p ) x − (cid:1) b (cid:16) (1 − p ) x +1 + ( x + 1) p (1 − p ) ( x +1) − (cid:17) | K |− b ≤ ((1 − p ) x + xp (1 − p ) x − ) b ≤ ((1 − p ) x + xp (1 − p ) x − ) h /x .
5t is also a routine calculation, that for any any p , 0 < p < t ≥ ((1 − p ) t + tp (1 − p ) t − ) /t = t =2 ((1 − p ) + 2 p (1 − p )) / . (2)Coming back to bounding p ′ , we have p ′ ≤ ((1 − p ) x + xp (1 − p ) x − ) h /x ≤ (2) ((1 − p ) + 2 p (1 − p )) h/ = (1 − p ) h/ ≤ e − p h/ . Observe also that the total number of hypergraphs H k is at most sr ≤ N r r ! ≤ (cid:18) Nr (cid:19) (1 + o (1)) . Thus, at least about a half of the possible r -sets of vertices belong to H k ’s that have at least twoedges, i.e., to H k ’s, k ∈ K . In other words, for sufficiently large N , h = X k ∈ K h k ≥ (cid:18) Nr (cid:19) − (cid:18) Nr (cid:19) (1 + o (1)) ≥ N r r ! . Recall that p = q √ ln N / √ N r − and q = √ r · r !. Then p ′ ≤ e − p h/ ≤ e − q (ln N/N r − ) N r / (12 r !) = e − ( q / r !) N ln N = e − (13 r/ N ln N ≤ N − rN . Also recall that s = ⌊ N r / (2 r · r !) ⌋ so,Pr[ s ( H ) ≤ s ] ≤ s N p ′ ≤ s N N − rN = o (1) . This implies that with high probability s ( H ) > s = cN r , for a constant c depending on r . Moreover,with high probability | E ( H ) | = Θ( p (cid:0) Nr (cid:1) ) = Θ( pN r ) = Θ( N ( r +1) / √ log N ).Our final lemma of this section upper-bounds the probability that a sum of i.i.d. uniformdiscrete random variables attains a particular value. We will use it as an ingredient in the upper6ound proof of Theorem 1. Lemma 3.
For any constant
C > , there exists ℓ such that for any ℓ > ℓ the following holds.There is an integer N = N ( ℓ ) such that for any N > N , if X , . . . , X ℓ are i.i.d. uniform discreterandom variables over [ N ] , then for any integer t , Pr[ X + · · · + X ℓ = t ] ≤ / ( e C N ) .Proof. We first establish the following claim, asserting the concavity of a sum of i.i.d. uniformdiscrete random variables.
Claim 1.
Let X , . . . , X ℓ be i.i.d. uniform discrete random variables over [ N ] and let X = X + · · · + X ℓ . Then for every real d ≥ it holds that Pr[ X = ( N + 1) ℓ/ − d ] = Pr[ X = ( N + 1) ℓ/ d ] ≥ Pr[ X = ( N + 1) ℓ/ d + 1] . Proof.
It will be slightly more convenient to define W i = X i − i = 1 , . . . , ℓ , W = W + · · · + W ℓ , q = N − W = qℓ/ − d ] = Pr[ W = qℓ/ d ] ≥ Pr[ W = qℓ/ d + 1] . Observe first that Pr[ W = qℓ/ − d ] = Pr[ W = qℓ/ d ] as W is symmetric around its mean qℓ/
2. Observe next that W ∈ { , . . . , qℓ } so the inequality is only interesting if d + 1 ≤ qℓ/ qℓ/ d is an integer. We prove the claim by induction on ℓ . The case ℓ = 1 trivially holds as W is uniform. Assuming the claim holds for ℓ −
1, we prove it for ℓ . Let W ∗ = W + · · · + W ℓ − so W = W ∗ + W ℓ . We prove that Pr[ W = qℓ/ d ] ≥ Pr[ W = qℓ/ d + 1]. Observe thatPr[ W = qℓ/ d ] = q X k =0 Pr[ W ∗ = qℓ/ d − k ] · Pr[ W ℓ = k ] = 1 q + 1 q X k =0 Pr[ W ∗ = qℓ/ d − k ] . Similarly, Pr[ W = qℓ/ d + 1] = 1 q + 1 q X k =0 Pr[ W ∗ = qℓ/ d + 1 − k ] . So, ( q + 1) (Pr[ W = qℓ/ d ] − Pr[ W = qℓ/ d + 1])= Pr[ W ∗ = qℓ/ d − q ] − Pr[ W ∗ = qℓ/ d + 1]= Pr[ W ∗ = q ( ℓ − / d − q/ − Pr[ W ∗ = q ( ℓ − / d + 1 + q/ ≥ | d − q/ | < | d + 1 + q/ | .To prove the lemma, let f = X + · · · + X ℓ and f = X ℓ +1 + · · · X ℓ .
7e estimate the probability Pr[ f + f = t ]. We will first need to prove two additional claims.The first is an anti-concentration result for f , f and the second is a concentration result for them.Throughout the remainder of the proof we assume that ℓ is sufficiently large as a function of C andthat N is sufficiently large as a function of ℓ . Claim 2.
Let j ∈ { , } . For any C > , there exists γ = γ ( C ) > such that for every real number x it holds that Pr[ x − γ √ ℓN ≤ f j ≤ x + γ √ ℓN ] ≤ e − C . (3) Proof.
Recall that each X i is uniform discrete over [ N ]. For the sake of our analysis it would beconvenient to obtain X i as follows. Let U i ∼ U [0 ,
1] (i.e. U i is uniform continuous in [0 , X i = ⌈ N U i ⌉ . Since Pr[ U i = 0] = 0, we have that X i is discrete uniform over [ N ] as the probabilitythat X i = t is 1 /N for each t ∈ [ N ]. Denote g = U + · · · + U ℓ and g = U ℓ +1 + · · · + U ℓ so we have f j − ℓ ≤ N g j ≤ f j for j ∈ { , } . Since ℓ < γ √ ℓN , it suffices to prove that for every real number y it holds that Pr[ y − γ √ ℓ ≤ g j ≤ y + 2 γ √ ℓ ] ≤ e − C . As g j is an Irwin-Hall distribution (i.e. the sum of i.i.d. copies of U [0 , ℓ/
2, themaximum of the left hand side is obtained when y = ℓ/ (cid:20) ℓ − γ √ ℓ ≤ g j ≤ ℓ γ √ ℓ (cid:21) ≤ e − C . (4)Since U i ∼ U [0 , / / √
12, so we have by the Central LimitTheorem thatlim ℓ →∞ Pr (cid:20) ℓ − γ √ ℓ ≤ g j ≤ ℓ γ √ ℓ (cid:21) = Φ(2 √ γ ) − Φ( − √ γ ) = 2Φ(2 √ γ ) − . Now, choose γ such that 2Φ(2 √ γ ) − e − C /
2. Then we havelim ℓ →∞ Pr (cid:20) ℓ − γ √ ℓ ≤ g j ≤ ℓ γ √ ℓ (cid:21) = 12 e C , implying that for all ℓ sufficiently large as a function of C we have that (4) holds. Claim 3.
Let j ∈ { , } . It holds that Pr[ | f j − ℓ ( N + 1) / | ≥ ℓ / N ] ≤ ℓ . (5) Proof.
As in the proof of the previous claim, since f j − ℓ ≤ N g j ≤ f j and since 2 ℓ ≤ ℓ / N , itsuffices to prove that Pr (cid:20) | g j − ℓ/ | ≥ ℓ / (cid:21) ≤ ℓ . Since g j is the sum of ℓ i.i.d. random variables, each in [0 , , it follows by8hernoff’s inequality (see, e.g. [2], Appendix A) thatPr (cid:20) | g i − ℓ/ | ≥ ℓ / (cid:21) ≤ e − ℓ / / (8 ℓ ) = 2 e − ℓ / / ≤ ℓ . Armed with the three claims we proceed as follows. Since f and f are independent and since ℓ ≤ f j ≤ ℓN we have that, for any t , 0 ≤ t ≤ ℓN ,Pr[ f + f = t ] = ℓN X k = ℓ Pr[ f = k ] · Pr[ f = t − k ] . We cover { ℓ, . . . , ℓN } with five (not necessarily disjoint) sets S , S , S , S , S defined as follows. S = { k | ℓ ( N + 1) / − γ √ ℓN ≤ k ≤ ℓ ( N + 1) / γ √ ℓN } S = { k | ℓ ( N + 1) / − γ √ ℓN ≤ t − k ≤ ℓ ( N + 1) / γ √ ℓN } S = { k | | k − ℓ ( N + 1) / | ≥ ℓ / N } S = { k | | ( t − k ) − ℓ ( N + 1) / | ≥ ℓ / N } S = { ℓ, . . . , ℓN } \ ( S ∪ S ∪ S ∪ S ) . For z ∈ { , , , , } let J z = P k ∈ S z Pr[ f = k ] · Pr[ f = t − k ] so that we havePr[ f + f = t ] ≤ J + J + J + J + J . We now bound each J z where we will use Claim 1, Claim 2, Claim 3, and the trivial boundPr[ f j = k ′ ] ≤ /N which holds for every k ′ ∈ [ N ] since f j is the sum of discrete random variables,each uniform on N possible values. By the definition of S and by Claim 2 applied to f with x = ℓ ( N + 1) / J = X k ∈ S Pr[ f = k ] · Pr[ f = t − k ] ≤ N X k ∈ S Pr[ f = k ] ≤ N e C . Similarly, by the definition of S and by Claim 2 applied to f with x = ℓ ( N + 1) / J = X k ∈ S Pr[ f = k ] · Pr[ f = t − k ] ≤ N X k ∈ S Pr[ f = t − k ] ≤ N e C . By the definition of S and by Claim 3 applied to f : J = X k ∈ S Pr[ f = k ] · Pr[ f = t − k ] ≤ N X k ∈ S Pr[ f = k ] ≤ N ℓ ≤ N e C .
9y the definition of S and by Claim 3 applied to f : J = X k ∈ S Pr[ f = k ] · Pr[ f = t − k ] ≤ N X k ∈ S Pr[ f = t − k ] ≤ N ℓ ≤ N e C . Finally consider J . To estimate it, we will distinguish between two cases, according to thevalue of t . Assume first that t ≤ ℓ ( N + 1) / t ≥ ℓ ( N + 1) /
3. In this case S ∪ S = { ℓ, . . . , ℓN } and hence S = ∅ implying that J = 0. Assume next that 2 ℓ ( N + 1) / < t < ℓ ( N + 1) /
3. First,observe that the number of elements of S is at most 2 ℓ / N + 1 < ℓ / N as it is disjoint from, say, S . Consider some term of J , namely Pr[ f = k ] · Pr[ f = t − k ] where k ∈ S . By Claim 1, we havethat Pr[ f = k ] ≤ Pr[ f = k ∗ ] where k ∗ ∈ S as k ∗ is closer to the mean ℓ ( N + 1) / k is. Butthe number of elements in S is at least 2 γ √ ℓN so we must have Pr[ f = k ] ≤ / | S | ≤ / (2 γ √ ℓN ).Similarly, by Claim 1, we have that Pr[ f = t − k ] ≤ Pr[ f = t − k ∗ ] where k ∗ ∈ S as t − k ∗ iscloser to the mean ℓ ( N + 1) / t − k is. But the number of elements in S is at least 2 γ √ ℓN so we must have Pr[ f = t − k ] ≤ / | S | ≤ / (2 γ √ ℓN ). Hence, in any case, J = X k ∈ S Pr[ f = k ] · Pr[ f = t − k ] ≤ ℓ / N · (cid:18) γ √ ℓN (cid:19) ≤ N e C . We have thus proved that Pr[ f + f = t ] ≤ e − C /N , as required. Let ǫ be given, 0 < ǫ <
1. Let n be sufficiently large and m be given such that n ≤ m ≤ n /ǫ .We shall construct a hypergraph H on n vertices and m hyperedges such that s ( H ) ≥ m − ǫ . Let r be a positive integer such that ǫ > / ( r + 1). Recall that Lemma 2 implies, for sufficiently large N and any positive δ , the existence of a hypergraph H ′ on N vertices and N ( r +1) / δ hyperedgessatisfying s ( H ′ ) ≥ cN r , for a constant c = c ( r ). Note that N ( r +1) / δ is slightly larger than theexpression for the number of hyperedges given in Lemma 2, but we can always add hyperedges ifnecessary as this does not decrease the parameter s . Next, we choose N such that m = N ( r +1) / δ .Thus H ′ has m hyperedges. Note that: N ≤ ( N ( r +1) / δ ) / ( r +1) = m / ( r +1) ≤ ( n /ǫ ) / ( r +1) ≤ n, so by just adding n − N isolated vertices to H ′ we obtain a hypergraph H with n vertices and m hyperedges and with s ( H ) = s ( H ′ ) ≥ cN r . Hence for δ sufficiently small s ( H ) ≥ cN r ≥ c ( m / ( r +1+2 δ ) ) r ≥ m − ǫ . We ignore rounding issues as these have no effect on the asymptotic statement of the theorem. .2 Proof of the upper bound of Theorem 1 Consider a hypergraph H = ( V, E ) on n vertices and m hyperedges. We shall argue that an ap-propriate random labeling is distinguishing with positive probability. Before we prove our upperbound m /C on s ( H ), we shall quickly remark that the upper bound s ( H ) ≤ m is easy to obtain.Indeed, to each vertex assign an integer value from [ m ] independently with probability 1 /m .Consider the probability p that two given distinct hyperedges e and e ′ get the same sum of thelabels. Fix an arbitrary vertex y in the symmetric difference of e and e ′ . Then assuming that allother labels in the union of e and e ′ are fixed, there is at most one value of the label assigned to y that makes the sum of labels in e and e ′ the same. Thus p ≤ /m . Taking the union bound overall (cid:0) m (cid:1) pairs of hyperedges, we see that the probability that the labeling is not distinguishing is atmost (cid:0) m (cid:1) /m < s ( n, m ) = o ( m ). This turns out to requiresignificantly more effort. We first describe the main idea of the proof. We consider a hypergraph H = ( V, E ) on n vertices and m hyperedges. Let C > N = ⌈ m /C ⌉ . Consider a labeling f : V ( H ) → [ N ] such that f ( v ) is assigned randomly with Pr[ f ( v ) = i ] = 1 /N for any i ∈ [ N ]and assignment of values to distinct vertices is independent. Let, for any set Q of vertices, s ( Q )denote P v ∈ Q f ( v ). For two hyperedges e, e ′ , let X ( e, e ′ ) = e \ e ′ . Observe that a vertex labeling f is distinguishing on H if for any two hyperedges s ( X ( e, e ′ )) = s ( X ( e ′ , e )). Let B ( e, e ′ ) be the (bad)event that s ( X ( e, e ′ )) = s ( X ( e ′ , e )).Consider sets D ( e, e ′ ) = X ( e, e ′ ) ∪ X ( e ′ , e ) and split the analysis into cases depending on thesize of D ( e, e ′ ). For small D ( e, e ′ ) we would like to apply the Lov´asz Local Lemma, but of coursethe lemma’s dependency digraph might have a high degree if there are vertices that belong to manysuch D ( e, e ′ )’s, called “dangerously popular” vertices. We treat them first observing that there arenot so many such vertices. Finally, we deal with large D ( e, e ′ )’s. For those we show that the badevent B ( e, e ′ ) does not happen by choosing a large set S of size 2 ℓ in X ( e, e ′ ) or in X ( e ′ , e ), fixingthe labels on the remaining vertices in D ( e, e ′ ) and showing that Pr[ B ( e, e ′ )] ≤ Pr[ s ( S ) = t ] fora specific value t , finally upper-bounding the latter using Lemma 3. We now proceed with thedetailed proof.For our fixed C , let K > P > C where K and P are positive integer constants chosen to satisfythe claimed inequalities used in the proof. They will only depend on C . For the rest of the proofwe assume that C > C , it holds for anysmaller positive value of C . • A pair of hyperedges e, e ′ is dangerous if | D ( e, e ′ ) | ≤ K . Otherwise, the pair is called non-dangerous . • We call a vertex w ∈ V ( H ) dangerously popular if for at least m /K dangerous pairs e, e ′ itholds that w ∈ D ( e, e ′ ). Let S be the set of all dangerously popular vertices. • For a pair e, e ′ ∈ E ( H ) (whether dangerous or not) let Y ( e, e ′ ) = X ( e, e ′ ) ∩ S , the set ofdangerously popular vertices in X ( e, e ′ ) and let Z ( e, e ′ ) = X ( e, e ′ ) \ Y ( e, e ′ ).11 We call a pair e, e ′ ∈ E ( H ) special if each vertex of D ( e, e ′ ) is dangerously popular, i.e. D ( e, e ′ ) = Y ( e, e ′ ) ∪ Y ( e ′ , e ). • Two special pairs e , e ′ and e , e ′ are equivalent if { X ( e , e ′ ) , X ( e ′ , e ) } = { X ( e , e ′ ) , X ( e ′ , e ) } .Observe that “equivalent” is an equivalence relation over the special pairs. • We call a non-dangerous and non-special pair e, e ′ ∈ E ( H ) newly dangerous if all but at most P vertices of D ( e, e ′ ) are dangerously popular (so 1 ≤ | Z ( e, e ′ ) ∪ Z ( e ′ , e ) | ≤ P for such pairs).We observe that that the number of dangerously popular vertices is | S | ≤ K . Indeed, the totalsum of cardinalities of all the D ( e, e ′ )’s ranging over all dangerous pairs is at most K (cid:0) m (cid:1) and as eachdangerously popular vertex is counted at least m /K times, there are at most K (cid:0) m (cid:1) / ( m /K ) ≤ K dangerously popular vertices.Recall that N = ⌈ m /C ⌉ . Our assignment of values from [ N ] to the vertices of H proceeds intwo steps. We will first assign values to the dangerously popular vertices such that some propertiesare guaranteed. We will then assign values to the remaining vertices. Step 1:
Assign random values to the dangerously popular vertices (i.e. the vertices in S ). As inthe proof of Lemma 3, for the purpose of our analysis, the random values are assigned as follows.Each w ∈ S is assigned uniformly and independently a random real g ( w ) in [0 , N ]. Then, we define f ( w ) = ⌈ g ( w ) ⌉ . Since Pr[ f ( w ) = 0] = 0, we have that f ( w ) is discrete uniform in [ N ] as theprobability that f ( w ) = t is 1 /N for each t ∈ [ N ].Recall that Y ( e, e ′ ) = X ( e, e ′ ) ∩ S . Let f ( e, e ′ ) = P w ∈ Y ( e,e ′ ) f ( w ). We say that Step 1 is successful if both of the following hold:1. For every special pair e, e ′ we have f ( e, e ′ ) = f ( e ′ , e ).2. For at most m e − C newly dangerous pairs e, e ′ it holds that | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N . Lemma 4.
With positive probability, Step 1 is successful.
Lemma 4 will be proved later, but for now assume that it holds, so fix an assignment of thevertices of S such that Step 1 is successful. Step 2:
Assign random values to the remaining n − | S | vertices. As in Step 1, we assign therandom values are follows. Each w ∈ V ( H ) \ S is assigned uniformly and independently a random real g ( w ) in [0 , N ]. Then, we define f ( w ) = ⌈ g ( w ) ⌉ . Recall that f ( w ) is discrete uniform in [ N ].This now defines for each hyperedge e ∈ E ( H ) the sum s ( e ) = P w ∈ e f ( w ). We need to estimatethe probability that s ( e ) = s ( e ′ ) for distinct hyperedges e, e ′ . We partition the pairs ( e, e ′ ) ofhyperedges into five types:(a) The special pairs.(b) The newly dangerous pairs for which | f ( e, e ′ ) − f ( e ′ , e ) | > P N .(c) The newly dangerous pairs for which | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N .(d) Non-dangerous pairs that are not newly dangerous and not special.12e) Dangerous pairs that are not special.We refer to these types by their letter. Each pair of hyperedges is of precisely one of these types.We now analyze each type. Let A a , A b , A c , A d , and A e be events that there is a pair e, e ′ of type(a), (b), (c), (d), or (e), respectively, such that s ( e ) = s ( e ′ ). We prove the following lemmas later. Lemma 5.
Pr[ A a ] = Pr[ A b ] = 0 . Lemma 6.
Pr[ A c ] ≤ e − C . Lemma 7.
Pr[ A d ] ≤ e − C . Lemma 8.
Pr[ A e ] ≤ − e − C . Lemmas 5, 6, 7, and 8 imply that Pr[ A a ∪ A b ∪ A c ∪ A d ∪ A e ] ≤ e − C + e − C + 1 − e − C < H . It remains to prove Lemmas 4, 5, 6, 7, and 8.In several proofs we shall need the following observation for any distinct subsets X and X ′ ofvertices, recalling that s ( X ) = P w ∈ X f ( w ),Pr( s ( X ) = s ( X ′ )) ≤ /N. (6)The reason for this observation to hold is the same as we outlined in the first paragraph of theproof - fixing all labels except for one vertex, say y , in the symmetric difference of X and X ′ , wesee that Pr( s ( X ) = s ( X ′ )) ≤ Pr( f ( y ) = t ) = 1 /N , for some specific value t . Proof of Lemma 5. If e, e ′ is a pair of type (a), then clearly s ( e ) − s ( e ′ ) = f ( e, e ′ ) − f ( e ′ , e ). Butsince Step 1 is successful, we have that f ( e, e ′ ) = f ( e ′ , e ) and hence s ( e ) = s ( e ′ ). Thus the event A a never happens.If e, e ′ is a pair of type (b), i.e., a newly-dangerous pair for which | f ( e, e ′ ) − f ( e ′ , e ) | > P N weproceed as follows. Assume without loss of generality that f ( e, e ′ ) − f ( e ′ , e ) > P N . Clearly s ( e ) = f ( e, e ′ ) + X w ∈ e ∩ e ′ f ( w ) + X w ∈ Z ( e,e ′ ) f ( w ) ≥ f ( e, e ′ ) + X w ∈ e ∩ e ′ f ( w ) . On the other hand, s ( e ′ ) = f ( e ′ , e ) + X w ∈ e ∩ e ′ f ( w ) + X w ∈ Z ( e ′ ,e ) f ( w ) ≤ f ( e ′ , e ) + X w ∈ e ∩ e ′ f ( w ) + P N, because | Z ( e ′ , e ) | ≤ P by the definition of newly-dangerous. It follows from the last two inequalitiesthat s ( e ) − s ( e ′ ) ≥ f ( e, e ′ ) − f ( e ′ , e ) − P N > , so we have that s ( e ) = s ( e ′ ). Thus the event A b never happens.13 roof of Lemma 6. Let e, e ′ be a pair of type (c), namely a newly dangerous pair for which it holdsthat | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N . As Step 1 is successful, we have that the number of pairs of type(c) is at most m e − C .By (6), we have that Pr[ s ( e ) = s ( e ′ )] ≤ /N . Since the number of pairs of type (c) is at most m e − C we have that Pr[ A c ] ≤ m e − C N = m e − C ⌈ m /C ⌉ ≤ Ce − C ≤ e − C . Proof of Lemma 8.
For a pair e, e ′ of type (e), let A ( e, e ′ ) be the event that s ( e ) = s ( e ′ ). Using(6) we have Pr[ A ( e, e ′ )] ≤ /N . Letting L denote the set of pairs of type (e), our goal is toprove that Pr[ ∩ { e,e ′ }∈ L A ( e, e ′ )] ≥ e − C as this is equivalent to proving that Pr[ A e ] ≤ − e − C .To this end, we will use the Lov´asz Local Lemma (LLL). Consider the dependency digraph onthe events A ( e, e ′ ) (note: there could be as many as (cid:0) m (cid:1) such events). We claim that any event A ( e, e ′ ) depends on not too many other events. Indeed, if Z ( e , e ′ ) ∪ Z ( e ′ , e ) is disjoint from Z ( e , e ′ ) ∪ Z ( e ′ , e ), then the event A ( e , e ′ ) is independent of the event A ( e , e ′ ) as they involveassignment of values to disjoint sets of vertices. Recall that the pairs of type (e) are, in particular,dangerous pairs. Hence | Z ( e, e ′ ) ∪ Z ( e ′ , e ) | ≤ K , for any pair e, e ′ of type (e). Furthermore, eachvertex of Z ( e, e ′ ) ∪ Z ( e ′ , e ) is not dangerously popular. Thus, we have that A ( e, e ′ ) is independent ofall but at most K · m /K = m /K other events. Denote { e , e ′ } ∼ { e , e ′ } if Z ( e , e ′ ) ∪ Z ( e ′ , e )is not disjoint from Z ( e , e ′ ) ∪ Z ( e ′ , e ). To apply LLL, define x ( e, e ′ ) = 2 /N . For any e , e oftype (e) it now holds that x ( e , e )Π { e ′ ,e ′ }∼{ e ,e } (1 − x ( e ′ , e ′ )) ≥ N (cid:18) − N (cid:19) m /K ≥ N (cid:18) − N (cid:19) CN/K > N ≥ Pr[ A ( e , e )]so the condition in the statement of LLL holds. So, by the LLL, we have thatPr[ ∩ { e,e ′ }∈ L A ( e, e ′ )] ≥ (1 − x ( e, e ′ )) | L | ≥ (cid:18) − N (cid:19) m / ≥ e − C , as required. Proof of Lemma 4.
We first prove that with probability at least 2 /
3, for every special pair e, e ′ wehave f ( e, e ′ ) = f ( e ′ , e ). Observe that the number of equivalence classes in the “equivalent” relationis at most 2 | S | | S | ≤ K (namely, a constant). Since for two equivalent special pairs e , e ′ and e , e ′ we have that f ( e , e ′ ) = f ( e ′ , e ) if and only if f ( e , e ′ ) = f ( e ′ , e ), it suffices to consider arepresentative special pair from every equivalence class. Now, if e, e ′ is a special pair then, using(6) we have that Pr[ f ( e, e ′ ) = f ( e ′ , e )] ≤ /N . We have by the union bound that the probability14hat for some special pair f ( e, e ′ ) = f ( e ′ , e ) is at most 4 K /N ≪ /
3. So, with probability at least2 /
3, for every special pair e, e ′ we have f ( e, e ′ ) = f ( e ′ , e ).We next prove that with probability at least 2 /
3, for at most m e − C newly dangerous pairs itholds that | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N (thus, we will have that Step 1 is successful with probabilityat least 1 − (1 − / − (1 − / >
0, as required). To prove this we will need to establish some“anti-concentration” result, and this will be possible by applying the law of large numbers to someappropriate random variable.Let us fix a newly dangerous pair u, v . We know that u, v is not a dangerous pair, namely | D ( e, e ′ ) | ≥ K . On the other hand, we know that D ( e, e ′ ) contains many dangerously popularvertices, since 1 ≤ | Z ( e, e ′ ) | ≤ P . So, either | Y ( e, e ′ ) | ≥ ( K − P ) / ≥ K/ | Y ( e ′ , e ) | ≥ ( K − P ) / ≥ K/
4. Assume without loss of generality that | Y ( e, e ′ ) | ≥ K/
4. Now, suppose we aregiven that f ( e ′ , e ) = t for some integer t . Given this information, we would like to upper bound theprobability that f ( e, e ′ ) lies in [ t − P N, t + P N ]. If we can provide an upper bound which does notdepend on t , then we have upper-bounded the probability that | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N regardlessof any given information.So, consider indeed the random variable f ( e, e ′ ). It is the sum of ℓ = | Y ( e, e ′ ) | ≥ K/ f ( e, e ′ ) = X + · · · + X ℓ where each X i is discrete uniform in [ N ]. Itwill be slightly more convenient to normalize as follows. Recall that each X i corresponds to some f ( w ) for w ∈ Y ( e, e ′ ) and that f ( w ) = ⌈ g ( w ) ⌉ . Hence X i is determined by first selecting uniformlyat random a real number W i in [0 , N ] and then setting X i = ⌈ W i ⌉ . Define U i = W i /N and noticethat U i ∼ U [0 ,
1] and that X i = ⌈ N U i ⌉ .Let g ( e, e ′ ) = U + · · · + U ℓ and observe that f ( e, e ′ ) − ℓ ≤ N g ( e, e ′ ) ≤ f ( e, e ′ ). Thus, it sufficesto upper bound the probability that g ( e, e ′ ) lies in [ t/N − P − ℓ/N, t/N + P ]. As ℓ/N ≤ K/N ≤ P ,it suffices to upper bound the probability that g ( e, e ′ ) lies in [ t ∗ − P, t ∗ + 2 P ] for some real number t ∗ . As U + · · · + U ℓ is an Irwin–Hall distribution which is concave in [0 , ℓ ], the latter probabilityis maximized when t ∗ = ℓ/
2, so it remains to upper bound the probability that g ( e, e ′ ) lies in[ ℓ/ − P, ℓ/ P ]. As the U i are i.i.d. each having mean and standard deviation 1 / √ g ( e, e ′ ), namely for every constant P lim ℓ →∞ Pr[ g ( e, e ′ ) ∈ [ ℓ/ − P, ℓ/ P ]] = 0 . This, in turn, means that for all K sufficiently large as a function of P, C (hence all ℓ sufficientlylarge since ℓ ≥ | K | / g ( e, e ′ ) ∈ [ ℓ/ − P, ℓ/ P ]] ≤ e − C . We have thus proved that Pr[ | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N ] ≤ e − C /
3. As there are less than m pairs toconsider, we have that the expected number of newly dangerous pairs satisfying | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N is at most m e − C /
3. By Markov’s inequality the probability that there are more than m e − C such pairs is less than 1 /
3, so indeed with probability at least 2 /
3, for at most m e − C newlydangerous pairs it holds that | f ( e, e ′ ) − f ( e ′ , e ) | ≤ P N .15 roof of Lemma 7.
Let e, e ′ be a pair of type (d), namely it is a non-dangerous pair and is not newlydangerous nor special. We will prove that Pr[ s ( e ) = s ( e ′ )] ≤ e − C /m . The lemma then followsas there are less than m such pairs to consider. Being non-dangerous and not newly dangerousmeans that | Z ( e, e ′ ) ∪ Z ( e ′ , e ) | ≥ P . Assume without loss of generality that | Z ( e, e ′ ) | ≥ P/
2. Let ℓ = ⌊ P/ ⌋ and let Z be a subset of Z ( e, e ′ ) of size 2 ℓ .Suppose we are given the value of f ( w ) for all w ∈ V ( H ) \ Z . Then, conditioned on thisinformation, for s ( e ) = s ( e ′ ) to hold, s ( Z ) must avoid a particular value t . By Lemma 3 we havethat Pr[ s ( Z ) = t ] ≤ / ( e C N ). Thus using the union bound over all pairs of hyperedges of type(d), we have that Pr[ A d ] ≤ m e C N ≤ Ce C ≤ e − C . Let G be a nonempty n -vertex graph with maximum degree ∆, minimum degree δ , and the largestnumber of vertices with pairwise distinct closed neighbourhoods equal to n ′ . We aim to show that n ′ + δ ∆ + 1 ≤ s ∗ ( G ) ≤ max { ( n − d ( v ) − d ( v ) + 1) + 2 : v ∈ V ( G ) } . For a vertex labeling f , we say that a pair of vertices u, v is bad if N [ u ] = N [ v ] and s ∗ f ( u ) = s ∗ f ( v ),otherwise the pair is good . Thus, a labeling is vertex sum-distinguishing if all pairs are good. Let ξ = max { ( n − d ( v ) − d ( v ) + 1) + 2 : v ∈ V ( G ) } , where d ( v ) is the degree of vertex v . Considera labeling f : V ( G ) → [ ξ ] with a smallest number of bad pairs. We argue that the number of badpairs is, in fact, zero.If not, let u, v be a bad pair. Let x = u if u and v are not adjacent and otherwise let x = w ,for some w ∈ ( N ( v ) \ N ( u )) ∪ ( N ( u ) \ N ( v )). Note that changing the label for x makes the pair u, v good. We shall change the label of x such that no good pair becomes bad, i.e., so that thenumber of bad pairs decreases. Denote the new labeling f ′ . Let t be a new value assigned to x ,i.e., t = f ( x ), f ′ ( x ) = t , f ′ ( z ) = f ( z ), for any z ∈ V ( G ) − x .We see that s ∗ f ′ ( y ) = s ∗ f ( y ) if y N [ x ] and s ∗ f ′ ( y ) = s ∗ f ( y ) − f ( x ) + t if y ∈ N [ x ]. Thus s ∗ f ′ ( y ) = s ∗ f ′ ( y ′ ) if s ∗ f ( y ) = s ∗ f ( y ′ ) and ( y, y ′ ∈ N [ x ] or y, y ′ ∈ V ( G ) − N [ x ]). We have that s ∗ f ′ ( y ) = s ∗ f ′ ( y ′ ) for y ∈ N [ x ] and y ′ N [ x ] if and only if s ∗ f ( y ) − f ( x ) + t = s ∗ f ( y ′ ). So, a new badpair can only appear if one vertex is in N [ x ] and another is not.Choose t ∈ Q , where Q = [ ξ ] \ (cid:0) { f ( x ) } ∪ { s ∗ f ( y ′ ) − s ∗ f ( y ) + f ( x ) : y ∈ N [ x ] , y ′ N [ x ] } (cid:1) . |{ s ∗ f ( y ′ ) − s ∗ f ( y ) + f ( x ) : y ∈ N [ x ] , y ′ N [ x ] }| ≤ |{ ( y, y ′ ) : y ∈ N [ x ] , y ′ N [ x ] }| = ( n − d ( x ) − d ( x ) + 1) ≤ ξ − , the set Q is non-empty, so there is a choice of t ≤ ξ , such that t = f ( x ) and s ∗ f ′ ( y ) = s ∗ f ′ ( y ′ ) forany y ∈ N [ x ] and y ′ N [ x ]. Since there is no bad pair y, y ′ for y, y ′ ∈ N [ x ] or y, y ′ N [ x ] in f ′ that was not bad in f and the pair u, v that was bad in f is no longer bad in f ′ , we see that thenumber of bad pairs in f ′ is strictly less than the number of bad pairs in f , a contradiction.For the lower bound, observe that if f is a vertex sum-distinguishing labeling of G with thelargest label k , then S = { s ∗ f ( v ) : v ∈ V ( G ) } ⊆ { ( δ ( G ) + 1) · , . . . , (∆ + 1) · k } . Since | S | ≥ n ′ , wehave n ′ ≤ | S | ≤ (∆ + 1) k − ( δ + 1) + 1, giving the desired lower bound.Note that the lower bound is tight for any pair δ, ∆, δ ≤ ∆. If δ = ∆ consider G = K ∆+1 , forwhich n ′ = 1, s ∗ ( G ) = 1, and ( n ′ + δ ) / (∆ + 1) = 1. If δ < ∆, consider G that is a vertex-disjointunion of K ∆+1 and K δ +1 . In this case n ′ = 2, s ∗ ( G ) = 1, and ⌈ ( n ′ + δ ) / (∆ + 1) ⌉ = 1. For a tree T and its vertex u , let L ( u ) be the set of leaves adjacent to u . Let L ( T ) = max {| L ( u ) | : u ∈ V ( T ) } . We shall prove for n ≥ T on n vertices, that s ∗ ( T ) ≤ n − − L ( T ) byinduction on n . Note that this bound is sharp for stars.The case n = 3 holds vacuously since T is a star. Suppose the statement holds for n ≥
3, andsuppose T has n + 1 vertices and is not a star. Choose a vertex u for which L ( T ) = | L ( u ) | , choosea leaf v adjacent to u , and let T ∗ = T − v . Then L ( T ) − | L ( u ) | − ≤ L ( T ∗ ). By inductionthere is a vertex sum-distinguishing labeling f : V ( T ∗ ) → [2 n − − L ( T ∗ )] of T ∗ . Observe that2 n − − L ( T ∗ ) ≤ n − − ( L ( T ) −
1) = 2 n − − L ( T ) < n − L ( T ) = 2( n + 1) − − L ( T ) . We define a labeling f ′ : V ( T ) → [2 n − L ( T )] such that f ′ ( u ) = f ( u ), u ∈ V ( T ∗ ), f ′ ( v ) = ξ .We argue that we can find an appropriate ξ so that the labeling f ′ does not contain bad pairs,i.e., pairs of vertices y, y ′ such that N [ y ] = N [ y ′ ] but s ∗ f ′ ( y ) = s ∗ f ′ ( y ′ ). Since there are no bad pairsin T ∗ under f , we have that y, y ′ is not a bad pair if y, y ′ ∈ T − { u, v } . Thus we need to consideronly the pairs y, y ′ , where y ∈ { u, v } . Let L = L ( u ) in T .If y = u and y ′ ∈ L , we see that s ∗ f ′ ( u ) > s ∗ f ′ ( y ′ ) regardless of ξ . Thus such a pair y, y ′ is notbad. A pair y = u, y ′ = u ′ , u ′ ∈ V ( T ∗ ) − ( { u } ∪ L ) can be bad if s ∗ f ′ ( u ) = s ∗ f ( u ) + ξ = s ∗ f ( u ′ ). Apair y = v, y ′ = u ′ , u ′ ∈ V ( T ∗ ) − u can be bad if s ∗ f ′ ( v ) = f ( u ) + ξ = s ∗ f ( u ′ ). Thus, if ξ X , where X = { s ∗ f ( u ′ ) − s ∗ f ( u ) : u ′ ∈ V ( T ∗ ) − ( { u } ∪ L ) } ∪ { s ∗ f ( u ′ ) − f ( u ) : u ′ ∈ V ( T ∗ ) − u } , then f ′ has no bad pairs on T . Note that | X | ≤ ( n − − | L | + ( n −
1) = 2 n − | L | −
2. Thus, thereis an available choice for ξ in [2 n − | L | ] − X . 17 .3 Proof of Theorem 4 Let H be the hypergraph from Lemma 2 and let H ∗ be the dual hypergraph of H , so irr ( H ∗ ) = s ( H ). Let n be the number of vertices of H ∗ that is the number of edges in H , i.e. n = | E ( H ) | =Θ( N ( r +1) / √ log N ). We also have that s ( H ) = cN r , so for ǫ < / ( r + 1) we have irr ( H ∗ ) ≥ cN r ≥ Cn r/ ( r +1) / log r n ≥ n − ǫ . As mentioned in the introduction, there are connections between the considered problem and Sidonsets. Recall that a B h [1]-Sidon set is a set X of integers such that for any integer q , there is atmost one subset X ′ of X , | X ′ | = h , so that the sum of elements from X ′ is q . The followingbounds on the sizes of Sidon sets are known: if X ⊆ [ K ] and X is a B h [1]-Sidon, then | X | ≤ ( h · h ! K ) /h (1 + o (1)), see for example [14, 15]. Let c ( h ) be a constant depending on h only suchthat | X | ≤ ( c ( h ) K ) /h (1+ o (1)) for any B h [1]-Sidon set X , X ⊆ [ K ]. Consider a hypergraph H thatis a union of a complete h -uniform hypergraph on N vertices and (cid:0) Nh (cid:1) − N isolated vertices. Then H has the same number n = N h /h !(1 + o (1)) of vertices and edges and s ( H ) ≥ (1 /c ( h )) h N h = Θ( n ).So, this only gives a linear lower bound on s ( n ) = s ( n, n ), much weaker than Theorem 1.Note that a similar problem defined on open neighbourhoods of the vertices of a graph isequivalent to the setting we considered on the complement G of the graph G . Indeed, if f is avertex sum-distinguishing labeling of G , then the numbers P u N [ v ] f ( u ), v ∈ V ( G ) are distinct forany two vertices with distinct open neighbourhoods. We see that V ( G ) − N [ v ] = N G ( v ), thus thesums considered correspond to the sums over open neighbourhoods in the complement.Yet another variant of s ( H ) is its restriction to injective labelings. Denoting the correspondingparameters by s inj ( H ) and s inj ( n, m ) we see, by definition, that s inj ( H ) ≥ s ( H ) so s inj ( n, m ) ≥ s ( n, m ). If a hypergraph H ′ is a union of H and all hypergedges consisting of exactly one vertex of H , then s inj ( H ) ≤ s ( H ′ ), and H ′ has at most | E ( H ) | + | V ( H ) | edges. Thus, Theorem 1 triviallyextends to s inj ( n, m ). By defining s ∗ inj ( n ) similarly and following the steps of Theorem 2, one canalso show that s ∗ inj ( G ) ≤ (∆ + 2) n , for any graph G on n vertices and maximum degree ∆.In this paper, we addressed hypergraphs on n vertices and m hyperedges, for n ≤ m ≤ n O (1) . Itmay be of some interest to determine the behavior of s ( n, m ) when m is larger than a polynomialfunction of n . As mentioned in the introduction, the closer m gets to 2 n , the closer s ( n, m ) gets tobe linear in m .Finally, it may be of some interest to improve the upper bound s ∗ ( G ) ≤ n (∆ + 1) given inTheorem 2 for regimes of ∆ that are significantly less than quadratic.18 eferences [1] Ahmad, A. and Al-Mushayt, O. and Baˇca, M., On edge irregularity strength of graphs. Appl.Math. Comput. 243 (2014), 607–610.[2] Alon, N. and Spencer J., The Probabilistic Method . John Wiley & Sons, 2004.[3] Axenovich, M., Harant, J., Przyby lo, J., Sot´ak, R., Voigt, M., Weidelich, J., A note on adjacentvertex distinguishing colorings of graphs. Discrete Appl. Math. 205 (2016), 1–7.[4] Balister, P., Bollob´as, B., Lehel, J., Morayne, M., Random hypergraph irregularity. SIAM J.Discrete Math. 30 (2016), no. 1, 465–473.[5] Bartnicki, T., Bosek, B., Czerwi´nski, S., Grytczuk, J., Matecki, G., ˙Zelazny, W., Additivecoloring of planar graphs. Graphs Combin. 30 (2014), no. 5, 1087–1098.[6] Bhattacharya, B., Das, S., Ganguly, S., Minimum-weight edge discriminators in hypergraphs.Electron. J. Combin. 21 (2014), no. 3, Paper 3.18, 19 pp.[7] Blokhuis, A., Sz˝onyi, T., Irregular weighting of 1-designs. Discrete Math. 131 (1994), no. 1-3,339–343.[8] Bollob´as, B. Pikhurko, O., Integer sets with prescribed pairwise differences being distinct.European J. Combin. 26 (2005), no. 5, 607–616.[9] Chartrand, G., Jacobson, M., Lehel, J., Oellermann, O., Ruiz, S., Saba, F., Irregular networks.250th Anniversary Conference on Graph Theory, Congr. Numer. 64 (1988), 197–210.[10] Gallian, J., A dynamic survey of graph labeling. Electron. J. Combin. 5 (1998), DynamicSurvey 6, 43 pp.[11] Guy, R., Sets of integers whose subsets have distinct sums. Theory and practice of combi-natorics, 141–154, North-Holland Math. Stud., 60, Ann. Discrete Math., 12, North-Holland,Amsterdam, 1982.[12] Gy´arf´as, A., Jacobson, M., Kinch, L., Lehel, J., Schelp, R., Irregularity strength of uniformhypergraphs. J. Combin. Math. Combin. Comput. 11 (1992), 161–172.[13] Nierhoff, T., A tight bound on the irregularity strength of graphs. SIAM J. Discrete Math. 13(2000), no. 3, 313–323.[14] Plagne, A., Recent progress on finite B h [ gg