DECOMPOSITIONS OF COMPLETE UNIFORM HYPERGRAPHS INTO HAMILTON BERGE CYCLES

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1 DECOMPOSITIONS OF COMPLETE UNIFORM HYPERGRAPHS INTO HAMILTON BERGE CYCLES DANIELA KÜHN AND DERYK OSTHUS Abstract. I 1973 Bermod, Germa, Heydema ad Sotteau cojectured that if divides (, the the complete -uiform hypergraph o vertices has a decompositio ito Hamilto Berge cycles. Here a Berge cycle cosists of a alteratig sequece v 1, e 1, v,..., v, e of distict vertices v i ad distict edges e i so that each e i cotais v i ad v i+1. So the divisibility coditio is clearly ecessary. I this ote, we prove that the cojecture holds wheever 4 ad 30. Our argumet is based o the Krusal-Katoa theorem. The case whe = 3 was already solved by Verrall, buildig o results of Bermod. 1. Itroductio A classical result of Waleci [1] states that the complete graph K o vertices has a Hamilto decompositio if ad oly if is odd. (A Hamilto decompositio of a graph G is a set of edge-disjoit Hamilto cycles cotaiig all edges of G. Aalogues of this result were proved for complete digraphs by Tillso [14] ad more recetly for (large touramets i [9]. Clearly, it is also atural to as for a hypergraph geeralisatio of Waleci s theorem. There are several otios of a hypergraph cycle, the earliest oe is due to Berge: A Berge cycle cosists of a alteratig sequece v 1, e 1, v,..., v, e of distict vertices v i ad distict edges e i so that each e i cotais v i ad v i+1. (Here v +1 := v 1 ad the edges e i are also allowed to cotai vertices outside {v 1,..., v }. A Berge cycle is a Hamilto (Berge cycle of a hypergraph G if {v 1,..., v } is the vertex set of G ad each e i is a edge of G. So a Hamilto Berge cycle has edges. Let K ( deote the complete -uiform hypergraph o vertices. Clearly, a ecessary coditio for the existece of a decompositio of K ( ito Hamilto Berge cycles is that divides (. Bermod, Germa, Heydema ad Sotteau [5] cojectured that this coditio is also sufficiet. For = 3, this cojecture follows by combiig the results of Bermod [4] ad Verrall [16]. We show that as log as is ot too small, the cojecture holds for 4 as well. Date: April 9, 014. The research leadig to these results was partially supported by the Europea Research Coucil uder the Europea Uio s Seveth Framewor Programme (FP/ / ERC Grat Agreemets o (D. Küh ad (D. Osthus. 1

2 DANIELA KÜHN AND DERYK OSTHUS Theorem 1. Suppose that 4 <, that 30 ad that divides (. The the complete -uiform hypergraph K ( o vertices has a decompositio ito Hamilto Berge cycles. Recetly, Peteci [13] cosidered a restricted type of decompositio ito Hamilto Berge cycles ad determied those for which K ( has such a restricted decompositio. Waleci s theorem has a atural extesio to the case whe is eve: i this case, oe ca show that K M has a Hamilto decompositio, wheever M is a perfect matchig. Similarly, the results of Bermod [4] ad Verrall [16] together imply that for all, either K (3 or K (3 M have a decompositio ito Hamilto Berge cycles. We prove a aalogue of this for 4. Note that Theorem immediately implies Theorem 1. Theorem. Let, N be such that 3 <. (i Suppose that 5 ad 0 or that = 4 ad 30. Let M be ay set cosistig of less tha edges of K ( such that divides E(K ( \ M. The K ( M has a decompositio ito Hamilto Berge cycles. (ii Suppose that = 3 ad 100. If ( 3 is ot divisible by, let M be ay perfect matchig i K (3, otherwise let M :=. The K (3 M has a decompositio ito Hamilto Berge cycles. Note that if is a prime ad ( is ot divisible by, the divides ad so i this case oe ca tae the set M i (i to be a uio of perfect matchigs. Also ote that (ii follows from the results of [4, 16]. However, our proof is far simpler, so we also iclude it i our argumet. Aother popular otio of a hypergraph cycle is the followig: a -uiform hypergraph C is a l-cycle if there exists a cyclic orderig of the vertices of C such that every edge of C cosists of cosecutive vertices ad such that every pair of cosecutive edges (i the atural orderig of the edges itersects i precisely l vertices. If l = 1, the C is called a tight cycle ad if l = 1, the C is called a loose cycle. We cojecture a aalogue of Theorem 1 for Hamilto l-cycles. Cojecture 3. For all, l N with l < there exists a iteger 0 such that the followig holds for all 0. Suppose that l divides ad that /( l divides ( (. The K has a decompositio ito Hamilto l-cycles. To see that the divisibility coditios are ecessary, ote that every Hamilto l-cycle cotais exactly /( l edges. Moreover, it is also worth otig the followig: cosider the umber N := l ( of cycles we require i the decompositio. The divisibility coditios esure that N is ot oly a iteger but also a multiple of f := ( l/h, where h is the highest commo factor of ad l. This is relevat as oe ca costruct a regular hypergraph from the edge-disjoit uio of t edge-disjoit Hamilto l-cycles if ad oly if t is a multiple of f.

3 DECOMPOSITIONS OF COMPLETE UNIFORM HYPERGRAPHS INTO BERGE CYCLES 3 The tight case l = 1 of Cojecture 3 was already formulated by Bailey ad Steves [1]. I fact, if ad are coprime, the case l = 1 already correspods to a cojecture made idepedetly by Barayai [3] ad Katoa o so-called wreath decompositios. A -partite aalogue of the tight case of Cojecture 3 was recetly proved by Schroeder [15]. Cojecture 3 is ow to hold approximately (with some additioal divisibility coditios o, i.e. oe ca fid a set of edge-disjoit Hamilto l-cycles which together cover almost all the edges of K (. This is a very special case of results i [, 6, 7] which guaratee approximate decompositios of quasi-radom uiform hypergraphs ito Hamilto l-cycles (agai, the proofs eed to satisfy additioal divisibility costraits.. Proof of Theorem Before we ca prove Theorem we eed to itroduce some otatio. Give itegers 0, we will write [] ( for the set cosistig of all -elemet subsets of [] := {1,..., }. The colexicographic order o [] ( is the order i which A < B if ad oly if the largest elemet of (A B \ (A B lies i B (for all distict A, B [] (. The lexicographic order o [] ( is the order i which A < B if ad oly if the smallest elemet of (A B \ (A B lies i A. Give l N with l ad a set S [] (, the lth lower shadow of S is the set l (S cosistig of all those t []( l for which there exists s S with t s. Similarly, give l N with + l ad a set S [] (, the lth upper shadow of S is the set + l (S cosistig of all those t [](+l for which there exists s S with s t. Give s R ad N, we write ( s := s(s 1 (s +1!. We eed the followig cosequece of the Krusal-Katoa theorem [8, 10]. Lemma 4. (i Let, N be such that 3. Give a oempty S [] (, defie s R by S = ( s. The (S ( s. (ii Suppose that S [] ( ad let c, d N {0} be such that c <, d < (c + 1 ad S = c ( c+1 + d. If 100 ad c 8 the + 1 (S c ( c + d/5. (iii If S [] ( ad S 1 the + (S S ( ( S 1 + S ( S 1. Proof. The Krusal-Katoa theorem states that the size of the lower shadow of a set S [] ( is miimized if S is a iitial segmet of [] ( i the colexicographic order. (i is a special case of a weaer (quatitative versio of this due to Lovász [11]. I order to prove (ii ad (iii, ote that wheever A, B [] ( the A < B i the colexicographic order if ad oly if [] \ A < [] \ B i the lexicographic order o [] ( with the order of the groud set reversed. Thus, by cosiderig complemets, it follows from the Krusal-Katoa theorem that the size of the upper shadow of a set S [] ( is miimized if S is a iitial segmet of [] ( i the lexicographic order. This immediately implies (iii. Moreover, if

4 4 DANIELA KÜHN AND DERYK OSTHUS S, c ad d are as i (ii, the ( ( ( 1 c 1 + S ( c c + 5 d, as required. + d( c ( d We will also use the followig result of Tillso [14] o Hamilto decompositios of complete digraphs. (The complete digraph DK o vertices has a directed edge xy betwee every ordered pair x y of vertices. So E(DK = ( 1. Theorem 5. The complete digraph DK o vertices has a Hamilto decompositio if ad oly if 4, 6. We are ow ready to prove Theorem. The strategy of the proof is as follows. Suppose for simplicity that l := ( /(( 1 is a iteger. (So i particular, the set M i Theorem is empty. Defie a auxiliary bipartite graph G with vertex classes A ad B of size ( ( as follows. Let A := E(K. Let B cosist of the edges of l copies D 1,..., D l of the complete digraph DK o vertices. G cotais a edge betwee z A ad xy B if ad oly if {x, y} z. It is easy to see that if G has a perfect matchig F, the K ( has a decompositio ito Hamilto Berge cycles. Ideed, for each i [l], choose a Hamilto decompositio Hi 1,..., H 1 i of D i (which exists by Theorem 5. The for all i [l] ad j [ 1], the set of all those edges of K ( which are mapped via F to the edges of H j i forms a Hamilto Berge cycle, ad all these cycles are edge-disjoit, as required. To prove the existece of the perfect matchig F, we use the Krusal-Katoa theorem to show that G satisfies Hall s coditio. Proof of Theorem. The first part of the proof for (i ad (ii is idetical. So let M be as i (i,(ii. (For (ii ote that if ( 3 is ot divisible by, the 3 divides ad divides ( 3 3. Let ( ( M M l( 1 l := ad m :=. ( 1 Note that m < 1 ad m N {0} sice divides ( M. Defie a auxiliary (balaced bipartite graph G with vertex classes A ad B of size ( M as follows. Let A := E(K ( ad A := A \ M. Let D 1,..., D l be copies of the complete digraph DK o vertices. For each i [l] let B i, B i be a partitio of E(D i such that for every pair xy, yx of opposite directed edges, B i cotais precisely oe of xy, yx. Apply Theorem 5 to fid m edge-disjoit Hamilto cycles H 1,..., H m i DK. We view the sets B 1,..., B l, B 1,..., B l ad E(H 1,..., E(H m as beig pairwise disjoit ad let B deote the uio of these sets. So B = A. Our auxiliary bipartite graph G cotais a edge betwee z A ad xy B if ad oly if {x, y} z.

5 DECOMPOSITIONS OF COMPLETE UNIFORM HYPERGRAPHS INTO BERGE CYCLES 5 We claim that G cotais a perfect matchig F. Before we prove this claim, let us show how it implies Theorem. For each i [l], apply Theorem 5 to obtai a Hamilto decompositio Hi 1,..., H 1 i of D i. For each i [l] ad each j [ 1] let A j i A be the eighbourhood of E(H j i i F. Note that each A j i is the edge set of a Hamilto Berge cycle of K ( M. Similarly, for each i [m] the eighbourhood A i of E(H i i F is the edge set of a Hamilto Berge cycle of K ( M. Sice all the sets A j i ad A i are pairwise disjoit, this gives a decompositio of K ( M ito Hamilto Berge cycles. Thus it remais to show that G satisfies Hall s coditio. So cosider ay oempty set S A ad defie s, a R with s ad 0 < a 1 by S = a ( ( = s. Defie b by NG (S B 1 = b (. Note that NG (S B 1 ( s by Lemma 4(i. But ( b s ( ( a s(s 1 (s + 1 ( ( ( s ( ( 1 ( + 1 s = 1, ad so b a /. Thus ( N G (S l N G (S B 1 la / = a / ( B E(H 1 E(H m a / ( A (. Let g := ( A + ( (. So if (1 a 1 / A ( ( ( = 1 g, the N G (S S. We ow distiguish three cases. Case Sice (( ( ( A ( 1 A A ( = (1 g (1 g, i this case (1 implies that N G (S S if S A ( 1. So suppose that S > A ( 1. Note that if 5 the every b B satisfies ( ( 5 6 N G (b M ( ( 1 sice + 3 ad 0. Hece N G (S = B.

6 6 DANIELA KÜHN AND DERYK OSTHUS So we may assume that = 4 ad S := B \N G (S. Thus S 1 := S B 1 ad l S (l + S 1. Note that N G(S 1 A \ S < ( 1. First suppose S 1 7. The ( 8 N G (S ( 8 M 7 ( > ( 1 by Lemma 4(iii ad our assumptio that 30. So we may assume that S 1 6. Apply Lemma 4(iii agai to see that ( ( 7 7 N G (S S 1 6( 7( 8 M l + S ( ( S S > S. (Here we use that 30 implies S l ( ( 3/4 4 > ad 6( 7( 8 ( (( ( 3/4. Thus N G (S S, as required. Case. = 3 Sice (( ( ( A 3( 1 A A 3( = (1 3g (1 g 3, i this case (1 implies that N G (S S if S A 3( 1. So suppose that S > A 3( 1 ad that S := B \ N G (S. Thus S 1 := S B 1 ad S (l + S 1 (( /3 + S 1. Let c, d N {0} be such that c <, d < (c+1 ad S 1 = c ( c+1 +d. Note that NG (S 1 A \S < 3( 1. Thus c < 8 sice otherwise ( ( 8 8 N G (S 1 8 M 8 3 > 3 ( > 3( 1 5 by Lemma 4(ii ad our assumptio that 100. (Here we use that ( 8 ( = ( ( Let M(S 1 deote the set of all those edges e M for which there is a pair xy S 1 with {x, y} e. Thus M(S 1 = + 1 (S 1 M. Recall that M is a matchig i the case whe = 3. Thus M(S 1 S 1. I particular M(S 1 d if c = 0. Apply Lemma 4(ii agai to see that ( c N G (S N G (S 1 c + 5 d M(S 1 4c ( { d /3 if c 1 d if c = 0 (c + d ( S S, 3

7 DECOMPOSITIONS OF COMPLETE UNIFORM HYPERGRAPHS INTO BERGE CYCLES 7 where we use that 100. (To see the fourth iequality, ote that if c 1 the ( 4c 5 3 4c ( 5 c 5 = 1c ( 10 3, whereas if c = 0 the d 5 d 11d Thus N G (S S, as required. Case 3. 1 If = 1 the K ( itself is a Hamilto Berge cycle, so there is othig to show. So suppose that =. I this case, it helps to be more careful with the choice of the Hamilto cycles H 1,..., H m : istead of applyig Theorem 5 to fid m edge-disjoit Hamilto cycles H 1,..., H m i DK, we proceed slightly differetly. Note first that l = 0. Suppose that is odd. The M = ad m = ( 1/. If is eve, the M = / ad m = / 1. I both cases we ca choose H 1,..., H m to be m edge-disjoit Hamilto cycles of K. The a perfect matchig i our auxiliary graph G still correspods to a decompositio of K ( M ito Hamilto Berge cycles. Also, i both cases E(H 1 E(H m cotais all but at most / distict elemets of [] (. Note that ( ( ( ( = ( ( 1 5 ( sice 0. Cosider ay b B. The ( ( N G (b M = Now cosider ay a A. The ( N G (a ( = M ( 3 ( 3 A. ( ( 3 3 B. So Hall s coditio is satisfied ad so G has a perfect matchig, as required. The lower bouds o have bee chose so as to streamlie the calculatios, ad could be improved by more careful calculatios. Refereces [1] R. Bailey ad B. Steves, Hamiltoia decompositios of complete -uiform hypergraphs, Discrete Math. 310 (010, [] D. Bal ad A. Frieze, Pacig tight Hamilto cycles i uiform hypergraphs, SIAM J. Discrete Math. 6 (01, [3] Zs. Barayai, The edge-colorig of complete hypergraphs I, J. Combi. Theory B 6 (1979, [4] J.C. Bermod. Hamiltoia decompositios of graphs, directed graphs ad hypergraphs, A. Discrete Math. 3 (1978, 1 8. [5] J.C. Bermod, A. Germa, M.C. Heydema, ad D. Sotteau, Hypergraphes hamiltoies, i Problèmes combiatoires et théorie des graphes (Colloq. Iterat. CNRS, Uiv. Orsay, Orsay, 1976, vol 60 of Colloq. Iterat. CNRS, Paris (1973, [6] A. Frieze ad M. Krivelevich, Pacig Hamilto cycles i radom ad pseudo-radom hypergraphs. Radom Structures ad Algorithms 41 (01, 1. [7] A. Frieze, M. Krivelevich ad P.-S. Loh, Pacig tight Hamilto cycles i 3-uiform hypergraphs, Radom Structures ad Algorithms 40 (01,

8 8 DANIELA KÜHN AND DERYK OSTHUS [8] G.O.H. Katoa, A theorem of fiite sets, i: Theory of Graphs, P. Erdős, G.O.H. Katoa Eds., Academic Press, New Yor, [9] D. Küh ad D. Osthus, Hamilto decompositios of regular expaders: a proof of Kelly s cojecture for large touramets, Adv. i Math. 37 (013, [10] J.B. Krusal, The umber of simplices i a complex, i: Mathematical Optimizatio Techiques, R. Bellma Ed., Uiversity of Califoria Press, Bereley, [11] L. Lovász, Combiatorial Problems ad Exercises, North-Hollad, Amsterdam, [1] E. Lucas, Récréatios Mathématiques, Vol., Gautheir-Villars, 189. [13] P. Peteci, O cyclic Hamiltoia decompositios of complete -uiform hypergraphs, Discrete Math. 35 (014, [14] T.W. Tillso, A Hamiltoia decompositio of K m, m 8, J. Combi. Theory B 9 (1980, [15] M.W. Schroeder, O Hamilto cycle decompositios of r-uiform r-partite hypergraphs, Discrete Math (014, 1 8. [16] H. Verrall, Hamilto decompositios of complete 3-uiform hypergraphs, Discrete Math. 13 (1994, Daiela Küh, Dery Osthus School of Mathematics Uiversity of Birmigham Edgbasto Birmigham B15 TT UK addresses: {d.uh,d.osthus}@bham.ac.u