Given non-negative integers $n_{i}$ and $\alpha_{i}$ with $0 \leq \alpha_{i}
\leq n_i$ $(i=1,2,...,k)$, an
$[\alpha_{1},\alpha_{2},...,\alpha_{k}]$-$k$-partite hypertournament on
$\sum_{1}^{k}n_{i}$ vertices is a $(k+1)$-tuple $(U_{1},U_{2},...,U_{k},E)$,
where $U_{i}$ are $k$ vertex sets with $|U_{i}|=n_{i}$, and $E$ is a set of
$\sum_{1}^{k}\alpha_{i}$-tuples of vertices, called arcs, with exactly
$\alpha_{i}$ vertices from $U_{i}$, such that any $\sum_{1}^{k}\alpha_{i}$
subset $\cup_{1}^{k}U_{i}^{\prime}$ of $\cup_{1}^{k}U_{i}$, $E$ contains
exactly one of the $(\sum_{1}^{k} \alpha_{i})!$ $\sum_{1}^{k}\alpha_{i}$-tuples
whose entries belong to $\cup_{1}^{k}U_{i}^{\prime}$. We obtain necessary and
sufficient conditions for $k$ lists of non-negative integers in non-decreasing
order to be the losing score lists and to be the score lists of some
$k$-partite hypertournament

Iványi, Antal

Pirzada, Shariefuddin

Zhou, Guofei

English

arXiv

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Acta Univ. Sapientiae, Informatica, 2, 2 (2010) 184–193Score lists in multipartitehypertournamentsShariefuddin PirzadaDepartment of Mathematics, Universityof Kashmir, India, and King FahdUniversity of Petroleum and Minerals,Saudi Arabiaemail: sdpirzada@yahoo.co.inGuofei ZhouDepartment of Mathematics, NanjingUniversity, Nanjing, Chinaemail: gfzhou@nju.edu.cnAntal Iva´nyiDepartment of Computer AlgebraEo¨tvo¨s Lora´nd University, Budapest, Hungaryemail: tony@inf.elte.huAbstract. Given non-negative integers ni and αi with 0 ≤ αi ≤ ni(i = 1, 2, . . . , k), an [α1, α2, . . . , αk]-k-partite hypertournament on∑k1 nivertices is a (k+ 1)-tuple (U1, U2, . . . , Uk, E), where Ui are k vertex setswith |Ui| = ni, and E is a set of∑k1 αi-tuples of vertices, called arcs, withexactly αi vertices from Ui, such that any∑k1 αi subset ∪k1U′i of ∪k1Ui, Econtains exactly one of the(∑k1 αi)!∑k1 αi-tuples whose entries belongto ∪k1U′i. We obtain necessary and sufficient conditions for k lists of non-negative integers in non-decreasing order to be the losing score lists andto be the score lists of some k-partite hypertournament.1 IntroductionHypergraphs are generalizations of graphs [1]. While edges of a graph are pairsof vertices of the graph, edges of a hypergraph are subsets of the vertex set,Computing Classification System 1998: G.2.2Mathematics Subject Classification 2010: 05C65Key words and phrases: hypergraph, hypertournament, multi hypertournament, score,losing score.184Score lists in multipartite hypertournaments 185consisting of at least two vertices. An edge consisting of k vertices is calleda k-edge. A k-hypergraph is a hypergraph all of whose edges are k-edges.A k-hypertournament is a complete k-hypergraph with each k-edge endowedwith an orientation, that is, a linear arrangement of the vertices contained inthe hyperedge. Instead of scores of vertices in a tournament, Zhou et al. [13]considered scores and losing scores of vertices in a k-hypertournament, andderived a result analogous to Landau’s theorem [6]. The score s(vi) or si of avertex vi is the number of arcs containing vi and in which vi is not the lastelement, and the losing score r(vi) or ri of a vertex vi is the number of arcscontaining vi and in which vi is the last element. The score sequence (losingscore sequence) is formed by listing the scores (losing scores) in non-decreasingorder.The following characterizations of score sequences and losing score sequencesin k-hypertournaments can be found in G. Zhou et al. [12].Theorem 1 Given two positive integers n and k with n ≥ k > 1, a non-decreasing sequence R = [r1, r2, . . . , rn] of non-negative integers is a losingscore sequence of some k-hypertournament if and only if for each j,j∑i=1ri ≥(jk),with equality when j = n.Theorem 2 Given positive integers n and k with n ≥ k > 1, a non-decreasingsequence S = [s1, s2, . . . , sn] of non-negative integers is a score sequence ofsome k-hypertournament if and only if for each j,j∑i=1si ≥ j(n− 1k− 1)+(n− jk)−(nk),with equality when j = n.Some recent work on the reconstruction of tournaments can be found inthe papers due to A. Iva´nyi [3, 4]. Some more results on k-hypertournamentscan be found in [2, 5, 9, 10, 11, 13]. The analogous results of Theorem 1 andTheorem 2 for [h, k]-bipartite hypertournaments can be found in [7] and for[α,β, γ]-tripartite hypertournaments in [8].Throughout this paper i takes values from 1 to k and ji takes values from 1to ni, unless otherwise stated.186 S. Pirzada, G. Zhou, A. Iva´nyiA k-partite hypergraph is a generalization of k-partite graph. Given non-negative integers ni and αi, (i = 1, 2, . . . , k) with ni ≥ αi ≥ 0 for each i, an[α1, α2, . . . , αk]-k-partite hypertournament (or briefly k-partite hypertourna-ment) M of order∑k1 ni consists of k vertex sets Ui with |Ui| = ni for each i,(1 ≤ i ≤ k) together with an arc set E, a set of ∑k1 αi-tuples of vertices, withexactly αi vertices from Ui, called arcs such that any∑k1 αi subset ∪k1U′i of∪k1Ui, E contains exactly one of the(∑k1 αi) ∑k1 αi-tuples whose αi entriesbelong to U′i.Let e = (u11, u12, . . . , u1α1 , u21, u22, . . . , u2α2 , . . . , uk1, uk2, . . . , ukαk), withuiji ∈ Ui for each i, (1 ≤ i ≤ k, 1 ≤ ji ≤ αi), be an arc in M and let h < t, welet e(u1h, u1t) denote to be the new arc obtained from e by interchanging u1hand u1t in e. An arc containing αi vertices from Ui for each i, (1 ≤ i ≤ k) iscalled an (α1, α2, . . . , αk)-arc.For a given vertex uiji ∈ Ui for each i, 1 ≤ i ≤ k and 1 ≤ ji ≤ αi, the scored+M(uiji) (or simply d+(uiji)) is the number of∑k1 αi-arcs containing uiji andin which uiji is not the last element. The losing score d−M(uiji) (or simplyd−(uiji)) is the number of∑k1 αi-arcs containing uiji and in which uiji is thelast element. By arranging the losing scores of each vertex set Ui separatelyin non-decreasing order, we get k lists called losing score lists of M and theseare denoted by Ri = [riji ]niji=1for each i, (1 ≤ i ≤ k). Similarly, by arrangingthe score lists of each vertex set Ui separately in non-decreasing order, we getk lists called score lists of M which are denoted as Si = [siji ]niji=1for each i(1 ≤ i ≤ k).2 Main resultsThe following two theorems are the main results.Theorem 3 Given k non-negative integers ni and k non-negative integersαi with 1 ≤ αi ≤ ni for each i (1 ≤ i ≤ k), the k non-decreasing listsRi = [riji ]niji=1of non-negative integers are the losing score lists of a k-partitehypertournament if and only if for each pi (1 ≤ i ≤ k) with pi ≤ ni,k∑i=1pi∑ji=1riji ≥k∏i=1(piαi), (1)with equality when pi = ni for each i (1 ≤ i ≤ k).Score lists in multipartite hypertournaments 187Theorem 4 Given k non-negative integers ni and k non-negative integersαi with 0 ≤ αi ≤ ni for each i (1 ≤ i ≤ k), the k non-decreasing listsSi = [siji ]niji=1of non-negative integers are the score lists of a k-partite hyper-tournament if and only if for each pi, (1 ≤ i ≤ k) with pi ≤ nik∑i=1pi∑ji=1siji ≥(k∑i=1αipini)(k∏i=1(niαi))+k∏i=1(ni − piαi)−k∏i=1(niαi), (2)with equality when pi = ni for each i (1 ≤ i ≤ k).We note that in a k-partite hypertournamentM, there are exactly∏ki=1(niαi)arcs and in each arc only one vertex is at the last entry. Therefore,k∑i=1ni∑ji=1d−M(uiji) =k∏i=1(niαi).In order to prove the above two theorems, we need the following Lemmas.Lemma 5 IfM is a k-partite hypertournament of order∑k1 ni with score listsSi = [siji ]niji=1for each i (1 ≤ i ≤ k), thenk∑i=1ni∑ji=1siji =[(k∑1=1αi)− 1]k∏i=1(niαi).Proof. We have ni ≥ αi for each i (1 ≤ i ≤ k). If riji is the losing score ofuiji ∈ Ui, thenk∑i=1ni∑ji=1riji =k∏i=1(niαi).The number of [αi]k1 arcs containing uiji ∈ Ui for each i, (1 ≤ i ≤ k), and1 ≤ ji ≤ ni isαinik∏t=1(ntαt).188 S. Pirzada, G. Zhou, A. Iva´nyiThus,k∑i=1ni∑ji=1siji =k∑i=1ni∑ji=1(αini) k∏1(ntαt)−(niαi)=(k∑i=1αi)k∏1(ntαt)−k∏1(niαi)=[(k∑1=1αi)− 1]k∏1(niαi).Lemma 6 If Ri = [riji ]niji=1(1 ≤ i ≤ k) are k losing score lists of a k-partitehypertournament M, then there exists some h with r1h < α1n1∏k1(npαp)so thatR′1 = [r11, r12, . . . , r1h+1, . . . , r1n1 ], R′s = [rs1, rs2, . . . , rst−1, . . . , rsns ] (2 ≤ s ≤k) and Ri = [riji ]niji=1, (2 ≤ i ≤ k), i 6= s are losing score lists of some k-partitehypertournament, t is the largest integer such that rs(t−1) < rst = . . . = rsns.Proof. Let Ri = [riji ]niji=1(1 ≤ i ≤ k) be losing score lists of a k-partite hyper-tournament M with vertex sets Ui = {ui1, ui2, . . . , uiji} so that d−(uiji) = rijifor each i (1 ≤ i ≤ k, 1 ≤ ji ≤ ni).Let h be the smallest integer such thatr11 = r12 = . . . = r1h < r1(h+1) ≤ . . . ≤ r1n1and t be the largest integer such thatrs1 ≤ rs2 ≤ . . . ≤ rs(t−1) < rst = . . . = rsnsNow, letR′1 = [r11, r12, . . . , r1h + 1, . . . , r1n1 ],R′s = [rs1, rs2, . . . , rst − 1, . . . , rsns(2 ≤ s ≤ k), and Ri = [riji ]niji=1, (2 ≤ i ≤ k), i 6= s.Clearly, R′1 and R′s are both in non-decreasing order.Since r1h < α1n1∏k1(npαp), there is at least one [αi]k1-arc e containing both u1hand ust with ust as the last element in e, let e′ = (u1h, ust). Clearly, R′1, R′sScore lists in multipartite hypertournaments 189and Ri = [riji ]niji=1for each i (2 ≤ i ≤ k), i 6= s are the k losing score lists ofM′ = (M− e) ∪ e′. The next observation follows from Lemma 6, and the proof can be easilyestablished.Lemma 7 Let Ri = [riji ]niji=1, (1 ≤ i ≤ k) be k non-decreasing sequences ofnon-negative integers satisfying (1). If r1n1 <α1n1∏k1(ntαt), then there exists sand t (2 ≤ s ≤ k), 1 ≤ t ≤ ns such that R′1 = [r11, r12, . . . , r1h + 1, . . . , r1n1 ],R′s = [rs1, rs2, . . . , rst−1, . . . , rsns ] and Ri = [riji ]niji=1, (2 ≤ i ≤ k), i 6= s satisfy(1).Proof of Theorem 3. Necessity. Let Ri, (1 ≤ i ≤ k) be the k losing scorelists of a k-partite hypertournament M(Ui, 1 ≤ i ≤ k). For any pi with αi≤ pi ≤ ni, let U′i = {uiji}piji=1(1 ≤ i ≤ k) be the sets of vertices such thatd−(uiji) = riji for each 1 ≤ ji ≤ pi, 1 ≤ i ≤ k. Let M′ be the k-partitehypertournament formed by U′i for each i (1 ≤ i ≤ k).Then,k∑i=1pi∑ji=1riji ≥k∑i=1pi∑ji=1d−M′(uiji)=k∏1(ptαt).Sufficiency. We induct on n1, keeping n2, . . . , nk fixed. For n1 = α1, theresult is obviously true. So, let n1 > α1, and similarly n2 > α2, . . . , nk > αk.Now,r1n1 =k∑i=1ni∑ji=1riji −n1−1∑j1=1r1j1 +k∑i=2ni∑ji=1riji≤k∏1(ntαt)−(n1 − 1α1) k∏2(ntαt)=[(n1α1)−(n1 − 1α1)] k∏2(ntαt)=(n1 − 1α1 − 1) k∏2(ntαt).190 S. Pirzada, G. Zhou, A. Iva´nyiWe consider the following two cases.Case 1. r1n1 =(n1 − 1α1 − 1)∏k2(ntαt). Then,n1−1∑j1=1r1j1 +k∑i=2ni∑ji=1riji =k∑i=1ni∑ji=1riji − r1n1=k∏1(ntαt)−(n1 − 1α1 − 1) k∏2(ntαt)=[(n1α1)−(n1 − 1α1 − 1)] k∏2(ntαt)=(n1 − 1α1) k∏2(ntαt).By induction hypothesis [r11, r12, . . . , r1(n1−1)], R2, . . . , Rk are losing scorelists of a k-partite hypertournament M′(U′1, U2, . . . , Uk) of order(∑ki=1 ni)−1. Construct a k-partite hypertournament M of order∑ki=1 ni as follows. InM′, let U′1 = {u11, u12, . . . , u1(n1−1)}, Ui = {uiji}niji=1for each i, (2 ≤ i ≤k). Adding a new vertex u1n1 to U′1, for each(∑ki=1 αi)-tuple containingu1n1 , arrange u1n1 on the last entry. Denote E1 to be the set of all these(n1 − 1α1 − 1)∏k2(ntαt) (∑ki=1 αi)-tuples. Let E(M) = E(M′) ∪ E1. Clearly,Ri for each i, (1 ≤ i ≤ k) are the k losing score lists of M.Case 2. r1n1 <(n1 − 1α1 − 1)∏k2(ntαt).Applying Lemma 7 repeatedly on R1 and keeping each Ri, (2 ≤ i ≤ k) fixeduntil we get a new non-decreasing list R′1 = [r′11, r′12, . . . , r′1n1] in which now′1n1=(n1 − 1α1 − 1)∏k2(ntαt). By Case 1, R′1, Ri (2 ≤ i ≤ k) are the losingscore lists of a k-partite hypertournament. Now, apply Lemma 6 on R′1, Ri(2 ≤ i ≤ k) repeatedly until we obtain the initial non-decreasing lists Ri foreach i (1 ≤ i ≤ k). Then by Lemma 6, Ri for each i (1 ≤ i ≤ k) are the losingscore lists of a k-partite hypertournament. Proof of Theorem 4. Let Si = [siji ]niji=1(1 ≤ i ≤ k) be the k score lists ofa k-partite hypertournament M(Ui, 1 ≤ i ≤ k), where Ui = {uiji}niji=1 withScore lists in multipartite hypertournaments 191d+M(uiji) = siji , for each i, (1 ≤ i ≤ k). Clearly,d+(uiji) + d−(uiji) =αini∏k1(ntαt), (1 ≤ i ≤ k, 1 ≤ ji ≤ ni).Let ri(ni+1−ji) = d−(uiji), (1 ≤ i ≤ k, 1 ≤ ji ≤ ni).Then Ri = [riji ]niji=1(i = 1, 2, . . . , k) are the k losing score lists of M. Con-versely, if Ri for each i (1 ≤ i ≤ k) are the losing score lists of M, thenSi for each i, (1 ≤ i ≤ k) are the score lists of M. Thus, it is enough toshow that conditions (1) and (2) are equivalent provided siji + ri(ni+1−ji) =(αini)∏k1(ntαt), for each i (1 ≤ i ≤ k and 1 ≤ ji ≤ ni).First assume (2) holds. Then,k∑i=1pi∑ji=1riji =k∑i=1pi∑ji=1(αini)( k∏1(ntαt))−k∑i=1pi∑ji=1si(ni+1−ji)=k∑i=1pi∑ji=1(αini)( k∏1(ntαt))− k∑i=1ni∑ji=1riji −k∑i=1ni−pi∑ji=1siji≥ k∑i=1pi∑ji=1(αini)( k∏1(ntαt))−[((k∑1αi)− 1)k∏1(niαi)]+k∑i=1(ni − pi)(αini) k∏1(ntαt)+k∏1(ni − (ni − pi)αi)−k∏1(niαi)=k∏1(niαi),with equality when pi = ni for each i (1 ≤ i ≤ k). Thus (1) holds.Now, when (1) holds, using a similar argument as above, we can show that(2) holds. This completes the proof. 192 S. Pirzada, G. Zhou, A. Iva´nyiAcknowledgementsThe research of the third author was supported by the European Union andthe European Social Fund under the grant agreement no. TA´MOP 4.2.1/B-09/1/KMR-2010-0003.References[1] C. Berge, Graphs and hypergraphs, translated from French by E. Minieka,North-Holland Mathematical Library 6, North-Holland Publishing Co.,Amsterdam, London, 1973. ⇒184[2] D. Brcanov, V. Petrovic, Kings in multipartite tournaments and hy-pertournaments, Numerical Analysis and Applied Mathematics: Interna-tional Conference on Numerical Analysis and Applied Mathematics 2009 :1, 2, AIP Conference Proceedings, 1168 (2009) 1255–1257. ⇒185[3] A. Iva´nyi Reconstruction of complete interval tournaments, Acta Univ.Sapientiae Inform., 1, 1 (2009) 71–88. ⇒185[4] A. Iva´nyi, Reconstruction of complete interval tournaments II, Acta Univ.Sapientiae Math., 2, 1 (2010) 47–71. ⇒185[5] Y. Koh, S. Ree, Score sequences of hypertournament matrices, On k-hypertournament matrices, Linear Algebra Appl., 373 (2003) 183–195.⇒185[6] H. G. Landau, On dominance relations and the structure of animal soci-eties. III. The condition for a score structure, Bull. Math. Biophys., 15(1953) 143–148. ⇒185[7] S. Pirzada, T. A. Chishti, T. A. Naikoo, Score lists in [h, k]-bipartitehypertournaments, Discrete Math. Appl., 19, 3 (2009) 321–328. ⇒185[8] S. Pirzada, T. A. Naikoo, G. Zhou, Score lists in tripartite hypertourna-ments, Graphs and Comb., 23, 4 (2007) 445–454. ⇒185[9] S. Pirzada G. Zhou, Score sequences in oriented k-hypergraphs, Eur. J.Pure Appl. Math., 1, 3 (2008) 10–20. ⇒185[10] S. Pirzada, G. Zhou, On k-hypertournament losing scores, Acta Univ.Sapientiae Inform., 2, 1 (2010) 5–9. ⇒185Score lists in multipartite hypertournaments 193[11] C. Wang, G. Zhou, Note on degree sequences of k-hypertournaments.Discrete Math., 308, 11 (2008) 2292–2296. ⇒185[12] G. Zhou, T. Yao, K. Zhang, On score sequences of k-tournaments. Euro-pean J. Comb., 21, 8 (2000) 993–1000. ⇒185[13] G. Zhou, On score sequences of k-tournaments. J. Appl. Math. Comput.,27 (2008) 149–158. ⇒185Received: June 23, 2010 • Revised: October 21, 2010

Score lists in multipartite hypertournaments

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