On decompositions of complete hypergraphs

Transcription

1 On decompositions of complete hypergraphs Sebastian M Cioabă 1, Department of Mathematical Sciences, University of Delaware, Newar, DE 19716, USA André Kündgen Department of Mathematics, California State University, San Marcos, CA , USA Jacques Verstraëte Department of Mathematics, University of California at San Diego, La Jolla, CA , USA Abstract We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon Key words: hypergraph, decomposition, Graham-Polla, Kneser graph, biclique 1 Introduction Given an r-uniform hypergraph H, let f r (H) denote the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edge set of Corresponding author addresses: cioaba@mathudeledu (Sebastian M Cioabă), aundgen@csusmedu (André Kündgen ), jverstra@mathucsdedu (Jacques Verstraëte) 1 Research supported by a startup grant from the Department of Mathematical Sciences of the University of Delaware Research supported by an Alfred P Sloan Research Fellowship and NSF Grant DMS Preprint submitted to Elsevier 3 September 010

2 H When r =, the parameter f (H) is also nown as the biclique partition number of the graph H and has been well studied (see [,9]) When H = K n, this parameter equals n 1 as shown by Graham and Polla [5] (see also [,10,1,14] for other proofs) In this note, we are interested in extending the theorem of Graham and Polla to complete r-uniform hypergraphs for larger values of r This seems a difficult and interesting extremal problem with connections to other areas such as theoretical computer science (the complexity of computing bilinear forms and symmetric polynomials, see [6,8,11]) and linear algebra (tensor ran computation of higher dimensional arrays, see [8]) To simplify our notation, let f r (n) = f r (K n (r) ), where K n (r) denotes the complete r-uniform hypergraph on n vertices In [1], Alon showed that f 3 (n) = n and that for n 4, f (n) ( (n ) ( ) ( ) ( ) ) n 1 n 3 n +1 ( ) (1) It was also proved in [1] by a recursive construction that r f r (n) f i ( n/ )f r i ( n/ ) () i=0 Using (), Alon showed that for fixed, as n f (n) n (1 + o(1))! In this note, we use nown results on biclique decomposition of the Kneser graphs to improve and simplify (1) and () We show that ( ) ( ) n ( ) f (n) (3) We expect that f (n) = ( ) n If true, this conjecture would imply that f +1 (n) = ( ) n 1 (see Theorem in the next section) The proofs of our results We follow Bollobás [3] for our hypergraph notation We use [n] to denote the set {1,, n} and [n] (r) to denote the family of all r-subsets of [n] If X 1,, X r are pairwise disjoint subsets of [n], we denote by r i=1 X i the complete r- partite r-uniform hypergraph whose parts are X 1,, X r The edges of this

3 hypergraph are all r-subsets Y of [n] such that Y X i = 1 for each i [r] When r =, i=1 X i is a complete bipartite subgraph (or biclique) of K n In this section, we present the proofs of our main results We use nown results about the biclique partition numbers of the Kneser graphs to prove the lower bound Recall that the Kneser graph K n: has vertex set [n] () with two -subsets being adjacent if and only if they are disjoint Theorem 1 For n, ) f (n) ( ( ) PROOF Let m = f (n) and consider a partition of the edge set of K n () into complete -partite -uniform hypergraphs H 1,, H m For i [m], let H i = j=1 A i j For each i [m], consider the following ( ) bicliques of the Kneser graph K n: For each partition X, Y of [] with X = Y =, we construct the biclique of K n: whose colour classes are the cartesian products j X A i j and l Y A i l We claim that these m ( ) bicliques partition the edge set of K n: Let xy be an edge of K n:, where x, y [n] () This means x y = and thus, x y is a -subset of [n] Because H 1,, H m partition the edge set of K n (), it follows that there exists an unique i [m] such that x y E(H i ) Now there exists an unique partition X, Y of [] such that X = Y =, x j X A i j and y l Y A i l This proves our claim and implies that f (n) f (K n: ) ( ) We show that f (K n: ) = ( ) This was proved by Vander Meulen [13] (see also [7]) For the sae of completness, we give a short proof here First, we can partition all the edges of K n: by stars centered at vertices x [n 1] () This is ( possible ) because [n 1] () is a vertex cover of K n: and implies f (K n: ) Recall that f (G) h(g), where h(g) is the maximum of the number of positive, and of the number of negative eigenvalues of the adjacency matrix of G (see [5,7] for more details) When G = K n:, it is nown (see [4] or [7]) that h(g) = ( ) This implies f (K n: ) ( ) and finishes the proof of the theorem 3

4 We obtain the upper bound for f (n) by a simple direct construction Theorem For each n + 1 3, we have that ( ) n 1 f (n 1) f +1 (n) PROOF We prove first that f +1 (n) ( ) n 1 For each -tuple 1 < i1 < i < i < n such that i j+1 i j > 1 for any 1 j 1, consider the complete ( + 1)-partite ( + 1)-uniform hypergraph H i1,,i whose parts are {1,, i 1 1}, {i 1 }, {i 1 + 1, i 1},, {i 1 + 1,, i 1}, {i } and {i + 1,, n} Note that the hypergraphs H i1,,i partition the edge set of K n (+1) This is because any edge j 1 j j +1 with 1 j 1 < j < < j +1 n is contained in precisely one of these hypergraphs, namely H j,,j ( Because ) 1 i 1 1 < < i n 1, it follows that there are n 1 such hypergraphs Hi1,,i This implies that f +1 (n) ( ) n 1 The inequality f (n 1) f +1 (n) is obvious and its proof is omitted (see [1] Lemma 1) Acnowledgments The first author thans Edwin van Dam for some useful comments Part of this wor was done while the first author held an NSERC PostDoctoral Fellowship at the University of California, San Diego and University of Toronto References [1] N Alon, Decomposition of the complete r-graph into complete r-partite r- graphs, Graphs and Combinatorics (1986), [] L Babai and P Franl, Linear Algebra Methods in Combinatorics, Department of Computer Science, The University of Chicago, September 199 [3] B Bollobás, Combinatorics Set systems, hypergraphs, families of vectors and combinatorial probability, Cambridge University Press, 1986 [4] C Godsil and G Royle, Algebraic Graph Theory, Springer 001 [5] RL Graham and HO Polla, On the addressing problem for loop switching, Bell Syst Tech J 50 (8) (1971),

5 [6] V Grolmusz, Computing elementary symmetric polynomials with a subpolynomial number of multiplications, SIAM J Comput 3 (003), [7] DA Gregory, B Heyin and KN Vander Meulen, Inertia and Biclique Decomposition of Joins of Graphs, Journal of Combin Theory, Series B 88 (001), [8] J Håstad, Tensor ran is NP-complete, J of Algorithms 11 (1990), [9] SD Monson, NJ Pullman and R Rees, A survey of clique and biclique coverings and factorizations of (0, 1)-matrices, Bull Inst Combin Appl 14 (1995), [10] G Pec, A new proof of a theorem of Graham and Polla, Discrete Math 49 (1984), [11] J Radharishnan, P Sen and S Vishwanathan, Depth-3 arithmetic for S n(x) and extensions of the Graham-Pollac theorem, FST TCS 000: Foundations of software technology and theoretical computer science (New Delhi), , Lecture Notes in Comput Sci, Springer, 000 [1] H Tverberg, On the decomposition of K n into complete bipartite graphs, J Graph Theory 6 (198), [13] KN Vander Meulen, Covers and Decompositions of Graphs by Complete Multipartite Subgraphs, PhD Thesis, Queen s University, Kingston 1995 [14] S Vishwanathan, A polynomial space proof of the Graham-Polla theorem, J Combin Theory Ser A 115 (008),