A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft

Cano Vila, María del Pilar

Perarnau Llobet, Guillem

Serra Albó, Oriol

Electronic Notes in Discrete Mathematics

English

UPCommons (Universitat Politècnica de Catalunya)

   UPCommons Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints   Aquesta és una còpia de la versió author’s final draft d'un article publicat a la revista [Electronic Notes in Discrete Mathematics]. URL d'aquest document a UPCommons E-prints: http://hdl.handle.net/2117/101772   Article publicat / Published paper:  Cano, M., Perarnau, G., Serra, O. Rainbow perfect matchings in r-partite graph structures. "Electronic notes in discrete mathematics", 3 Octubre 2016, vol. 54, p. 193-198. Doi: 10.1016/j.endm.2016.09.034  Rainbow perfect matchings in r-partite graphstructuresPilar Cano 1Department of MathematicsUniversitat Politècnica de CatalunyaBarcelona, SpainGuillem Perarnau 2School of MathematicsUniversity of BirminghamBirmingham, B15 2TT, UKOriol Serra 3Department of MathematicsUniversitat Politècnica de Catalunya, andBarcelona Graduate School of Mathematics, Barcelona, SpainAbstractA matchingM in an edge–colored (hyper)graph is rainbow if each pair of edges inMhave distinct colors. We extend the result of Erdős and Spencer on the existence ofrainbow perfect matchings in the complete bipartite graph Kn,n to complete bipar-tite multigraphs, dense regular bipartite graphs and complete r-partite r-uniformhypergraphs. The proof of the results use the Lopsided version of the Local LovászLemma.Keywords: Rainbow matchings, Lovász Local lemma.© <2016>. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ 1 IntroductionA latin transversal in a square matrix of order n is a set of entries, no two in thesame row or column, which are pairwise distinct. A longstanding conjectureof Ryser states that every Latin square with odd order has a latin transversal.Some results on the existence of a large partial latin transversal can be foundin [11,6,16]. Mainly motivated by Ryser’s conjecture, Erdős and Spencer [8]proved the following result:Theorem 1.1 [8] Let A be a square matrix of order n in which no entryappears more thann 14e times. Then A has a latin transversal.A matching M in an edge–colored graph is rainbow if no two edges inM have the same color. Theorem 1.1 can be rephrased by saying that everyedge–coloring of the complete graph Kn,n in which no color appears more than(n  1)/4e times contains a rainbow perfect matching.There has been a considerable amount of literature related to the study ofexistence and enumeration of rainbow matchings in graphs and hypergraphs.Bissacot et al. [5] improved the constant 14e in Theorem 1.1 to27256 and theirresult was made algorithmic in [10]. Perarnau and Serra [14] provided boundson the number of latin transversals under the hypothesis of Theorem 1.1.Here we are interested in analogous versions of Theorem 1.1 for other graphstructures such as complete bipartite multigraphs, regular bipartite graphsand hypergraphs.1.1 Complete multigraphs:Aharoni and Berger [2] considered the problem of finding a rainbow matchingfrom a slightly di↵erent perspective by asking how large a given collectionM of matchings (not necessarily edge–disjoint) should be to ensure that amatching of some given size t can be obtained by using an edge of each ofthe matchings in M. If every matching in M is given a di↵erent color, thenwe aim to find a rainbow matching of size t. The question, posed in thegeneral context of r–partite r–uniform hypergraphs, was connected by Alon[4] to zero–sum problems in additive combinatorics by disproving a conjecturein [2] and providing some bounds on the size of M in terms of t. Glebov,Sudakov and Szabó [9] improved the upper bound in [4] from superexponential1 Email: pilukno@gmail.com2 Email: p.melliug@gmail.com3 Email: oriol.serra@upc.eduto polynomial in t. In the case of bipartite graphs (r = 2), Clemens andEhrenmüller showed that |M|   3t/2 + o(t) su ces [7].Nevertheless, the main obstruction for a family M of matchings to containa rainbow matching seems to be the existence of edges appearing in a largenumber of the matchings. This motivates the study of the existence of rainbowmatchings in multigraphs with bounded multiplicity of edges. A first step inthis direction in the context of Theorem 1.1 is the following result.Theorem 1.2 For every n   1 and m   1, let G be an edge–colored completebipartite mutligraph with both stable sets of cardinality n and where each edgehas multiplicity m. If each color appears less than m(n 1)4e times, then G has arainbow perfect matching.1.2 Regular bipartite graphs:Another direction of research considers general graphs other than the completebipartite graph. The minimum colour degree is the smallest number of distinctcolours on the edges incident with a vertex. Kostochka and Yancey [12] showedthat every edge-coloured graph on n vertices with minimum colour degree atleast k contains a rainbow matching of size at least k, provided n   (17/4)k2,and this bound on n was reduced to n   4k   4 by Lo and Tan [13].Hall’s theorem ensures that d–regular bipartite graphs have a perfect match-ing. Thus, given an edge–colored d–regular bipartite graph, one can aim atfinding a perfect rainbow matching. An answer to that question under thesetting of Theorem 1.1 and for relatively large d is given by following result.Theorem 1.3 For every n   1 and d   n/2 + 2, let G be an edge–coloredd-regular bipartite graph with both stable sets of cardinality n. If each colorappears less than(2d n 3)24ed times, then G has a rainbow perfect matching.1.3 r–partite r–uniform hypergraphs:A rainbow matching in a bipartite graph can be understood as a matchingin a certain 3–partite 3–uniform hypergraph by adding to each edge a thirdvertex identifying its color. The existence of matchings in r–partite r–uniformhypergraphs has also been intensively studied. A conjecture of Ryser relatingthe matching number of such an r–uniform hypergraph with the coveringnumber was solved by Aharoni [1] for r = 3 by using a hypergraph version ofHall’s theorem obtained by Aharoni and Haxel [3]. However, direct applicationof this result only guarantees the existence of a relatively small rainbow partialmatching in the corresponding hypergraph. Our last result concerns rainbowmatchings in complete r–partite r–uniform hypergraphs.Theorem 1.4 For every n   1 and r   2, let K(r)n,...,n be an edge–coloredr–partite r–uniform hypergraph where each stable set has cardinality n. Ifeach color appears less than(n 1)r 12re times, then K(r)n,...,n has a rainbow perfectmatching.2 Lopsided Local Lovász LemmaThe three above theorems are proved by using a Lopsided version of the LocalLovász Lemma. This version was introduced by Erdős and Spencer [8] pre-cisely to prove Theorem 1.1 and has become a powerful tool in probabilisticcombinatorics. We use the following version of the result.Let E be a family of events in a probability space and p 2 (0, 1). A graphH which has the events as vertices is a p–lopsided graph for E if, for each eventA 2 E , and every subset S ✓ E \NH(A),P[A| \B2S B]  p.Lemma 2.1 (LLLL) Let E be a family of events in a probability space andlet H be a p–lopsided graph for E . Let d be the maximum degree of H. Ifep(d+ 1)  1,thenP⇥\A2EA⇤> 0.The proofs of Theorems 1.2, 1.3 and 1.4 follow the same strategy. Weconsider a matching M chosen uniformly at random and a set of events Econsisting of pairs of edges in M with the same color. Therefore, if none ofthese events occur, M is a rainbow matching. The conclusion of the theoremsfollows from an application of the LLLL, once an appropriate p–lopsided graph,for a suitable chosen p, is defined. The construction of the p–lopsided graphis analogous in all the cases, but their analysis di↵ers considerably in each ofthem and uses an elegant switching argument.In the case of Theorem 1.2 we follow the same proof strategy as the onein [8]. However, it is not clear to us that a uniform bound on the multiplicityof edges can be treated with the same tools.The case of a d-regular bipartite graph G requires a more careful analysis,since the uniform probability space on the perfect matchings of G is not aproduct space as for complete graphs. We believe that it is possible to extendour result to sparser regular bipartite graphs, provided that every color doesnot appear too many times on the edges of the graph.Finally, the case of complete r–partite r–uniform hypergraphs is techni-cally more involved due to the higher degree of complexity of the hypergraphstructure, and all steps in the proof must be adapted to this setting.References[1] R. Aharoni, Ryser’s conjecture for tripartite 3-graphs. Combinatorica 21 (1)(2001) 1–4.[2] R. Aharoni and E. Berger, Rainbow matchings in r-partite graphs. ElectronicJ. Combinatorics, 2009.[3] R. Aharoni and P. Haxell, Hall’s theorem for hypergraphs. J. Graph Theory 35(2000), 83–88.[4] N. Alon, Multicolored matchings in hypergraphs, Moscow Journal ofCombinatorics and Number Theory 1 (2011) 3–10.[5] R. Bissacot, R. Fernández, A. Procacci and B. Scoppola, An improvement ofthe Lovász Local Lemma via cluster expansion. Combinatorics, Probability andComputing 20(5) (2011): 709–719.[6] A. E. Brouwer, A. J. de Vries and R. M. Wieringa, A lower bound for thelength of partial transversals in a Latin square. Nieuw Archief Voor Wiskunde,26 (1978), 330–332.[7] D. Clemens and J. Ehrenmüller, An improved bound on the sizes of matchingsguaranteeing a rainbow matching. arXiv:1503.00438v1.[8] P. Erdös and J. Spencer, Lopsided Lovász local lemma and Latin transversals.Discrete Appl. Math 30 (1991), no. 2-3, 151–154, ARIDAM III (New Brunswick,NJ, 1988).[9] R. Glebov, B. Sudakov and T. Szabó, How many colors guarantee a Rainbowmatching?. Electronic J. Combinatorics 21 (1) (2014) 1–27.[10] D.G. Harris and A. Srinivasan. A constructive algorithm for the Lovsz LocalLemma on permutations. Proceedings of the Twenty-Fifth Annual ACM-SIAMSymposium on Discrete Algorithms. SIAM, 2014.[11] P. Hatami and P. W. Shor, A lower bound for the length of a partial transversalin a Latin square. J Combinatorics, Series A 115 (7) (2008) 1103–1113.[12] A. Kostochka and M. Yancey, Large rainbow matchings in edge-coloured graphs,Combinatorics, Probability and Computing 21 (2012), 255–263.[13] A. Lo, T.S. Tan, A note on large rainbow matchings in edge-coloured graphs,Graphs and Combinatorics 30, 2, (2014) 389–393.[14] G. Perarnau and O. Serra, Rainbow perfect matchings in complete bipartitegraphs: existence and counting. Combinatorics, Probability and Computing 22(5) (2013) 783–799.[15] H.J. Ryser, Neuere Probleme der Kombinatorik. Vortrogeber Kombinatorik,Oberwolfach, Matematisches Forschungsinstitute Obersolfach, 28–29 July 1967,pp. 69–91.[16] D. E. Woolbright, An n⇥n latin square has a transversal with at least n pndistinct symbols. J. Combinatorial Theory, Series A 24 (2) (1978) 235–237.

Rainbow perfect matchings in r-partite graph structures

UPCommons. Portal del coneixement obert de la UPC

   UPCommons Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints   Aquesta és una còpia de la versió author’s final draft d'un article publicat a la revista [Electronic Notes in Discrete Mathematics]. URL d'aquest document a UPCommons E-prints: http://hdl.handle.net/2117/101772   Article publicat / Published paper:  Cano, M., Perarnau, G., Serra, O. Rainbow perfect matchings in r-partite graph structures. "Electronic notes in discrete mathematics", 3 Octubre 2016, vol. 54, p. 193-198. Doi: 10.1016/j.endm.2016.09.034  Rainbow perfect matchings in r-partite graphstructuresPilar Cano 1Department of MathematicsUniversitat Polite`cnica de CatalunyaBarcelona, SpainGuillem Perarnau 2School of MathematicsUniversity of BirminghamBirmingham, B15 2TT, UKOriol Serra 3Department of MathematicsUniversitat Polite`cnica de Catalunya, andBarcelona Graduate School of Mathematics, Barcelona, SpainAbstractA matchingM in an edge–colored (hyper)graph is rainbow if each pair of edges inMhave distinct colors. We extend the result of Erdo˝s and Spencer on the existence ofrainbow perfect matchings in the complete bipartite graph Kn,n to complete bipar-tite multigraphs, dense regular bipartite graphs and complete r-partite r-uniformhypergraphs. The proof of the results use the Lopsided version of the Local Lova´szLemma.Keywords: Rainbow matchings, Lova´sz Local lemma.© <2016>. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ 1 IntroductionA latin transversal in a square matrix of order n is a set of entries, no two in thesame row or column, which are pairwise distinct. A longstanding conjectureof Ryser states that every Latin square with odd order has a latin transversal.Some results on the existence of a large partial latin transversal can be foundin [11,6,16]. Mainly motivated by Ryser’s conjecture, Erdo˝s and Spencer [8]proved the following result:Theorem 1.1 [8] Let A be a square matrix of order n in which no entryappears more than n 14e times. Then A has a latin transversal.A matching M in an edge–colored graph is rainbow if no two edges inM have the same color. Theorem 1.1 can be rephrased by saying that everyedge–coloring of the complete graph Kn,n in which no color appears more than(n  1)/4e times contains a rainbow perfect matching.There has been a considerable amount of literature related to the study ofexistence and enumeration of rainbow matchings in graphs and hypergraphs.Bissacot et al. [5] improved the constant 14e in Theorem 1.1 to27256 and theirresult was made algorithmic in [10]. Perarnau and Serra [14] provided boundson the number of latin transversals under the hypothesis of Theorem 1.1.Here we are interested in analogous versions of Theorem 1.1 for other graphstructures such as complete bipartite multigraphs, regular bipartite graphsand hypergraphs.1.1 Complete multigraphs:Aharoni and Berger [2] considered the problem of finding a rainbow matchingfrom a slightly di↵erent perspective by asking how large a given collectionM of matchings (not necessarily edge–disjoint) should be to ensure that amatching of some given size t can be obtained by using an edge of each ofthe matchings in M. If every matching in M is given a di↵erent color, thenwe aim to find a rainbow matching of size t. The question, posed in thegeneral context of r–partite r–uniform hypergraphs, was connected by Alon[4] to zero–sum problems in additive combinatorics by disproving a conjecturein [2] and providing some bounds on the size of M in terms of t. Glebov,Sudakov and Szabo´ [9] improved the upper bound in [4] from superexponential1 Email: pilukno@gmail.com2 Email: p.melliug@gmail.com3 Email: oriol.serra@upc.eduto polynomial in t. In the case of bipartite graphs (r = 2), Clemens andEhrenmu¨ller showed that |M|   3t/2 + o(t) su ces [7].Nevertheless, the main obstruction for a familyM of matchings to containa rainbow matching seems to be the existence of edges appearing in a largenumber of the matchings. This motivates the study of the existence of rainbowmatchings in multigraphs with bounded multiplicity of edges. A first step inthis direction in the context of Theorem 1.1 is the following result.Theorem 1.2 For every n   1 and m   1, let G be an edge–colored completebipartite mutligraph with both stable sets of cardinality n and where each edgehas multiplicity m. If each color appears less than m(n 1)4e times, then G has arainbow perfect matching.1.2 Regular bipartite graphs:Another direction of research considers general graphs other than the completebipartite graph. The minimum colour degree is the smallest number of distinctcolours on the edges incident with a vertex. Kostochka and Yancey [12] showedthat every edge-coloured graph on n vertices with minimum colour degree atleast k contains a rainbow matching of size at least k, provided n   (17/4)k2,and this bound on n was reduced to n   4k   4 by Lo and Tan [13].Hall’s theorem ensures that d–regular bipartite graphs have a perfect match-ing. Thus, given an edge–colored d–regular bipartite graph, one can aim atfinding a perfect rainbow matching. An answer to that question under thesetting of Theorem 1.1 and for relatively large d is given by following result.Theorem 1.3 For every n   1 and d   n/2 + 2, let G be an edge–coloredd-regular bipartite graph with both stable sets of cardinality n. If each colorappears less than (2d n 3)24ed times, then G has a rainbow perfect matching.1.3 r–partite r–uniform hypergraphs:A rainbow matching in a bipartite graph can be understood as a matchingin a certain 3–partite 3–uniform hypergraph by adding to each edge a thirdvertex identifying its color. The existence of matchings in r–partite r–uniformhypergraphs has also been intensively studied. A conjecture of Ryser relatingthe matching number of such an r–uniform hypergraph with the coveringnumber was solved by Aharoni [1] for r = 3 by using a hypergraph version ofHall’s theorem obtained by Aharoni and Haxel [3]. However, direct applicationof this result only guarantees the existence of a relatively small rainbow partialmatching in the corresponding hypergraph. Our last result concerns rainbowmatchings in complete r–partite r–uniform hypergraphs.Theorem 1.4 For every n   1 and r   2, let K(r)n,...,n be an edge–coloredr–partite r–uniform hypergraph where each stable set has cardinality n. Ifeach color appears less than (n 1)r 12re times, then K(r)n,...,n has a rainbow perfectmatching.2 Lopsided Local Lova´sz LemmaThe three above theorems are proved by using a Lopsided version of the LocalLova´sz Lemma. This version was introduced by Erdo˝s and Spencer [8] pre-cisely to prove Theorem 1.1 and has become a powerful tool in probabilisticcombinatorics. We use the following version of the result.Let E be a family of events in a probability space and p 2 (0, 1). A graphH which has the events as vertices is a p–lopsided graph for E if, for each eventA 2 E , and every subset S ✓ E \NH(A),P[A| \B2S B]  p.Lemma 2.1 (LLLL) Let E be a family of events in a probability space andlet H be a p–lopsided graph for E . Let d be the maximum degree of H. Ifep(d+ 1)  1,thenP⇥\A2EA⇤ > 0.The proofs of Theorems 1.2, 1.3 and 1.4 follow the same strategy. Weconsider a matching M chosen uniformly at random and a set of events Econsisting of pairs of edges in M with the same color. Therefore, if none ofthese events occur, M is a rainbow matching. The conclusion of the theoremsfollows from an application of the LLLL, once an appropriate p–lopsided graph,for a suitable chosen p, is defined. The construction of the p–lopsided graphis analogous in all the cases, but their analysis di↵ers considerably in each ofthem and uses an elegant switching argument.In the case of Theorem 1.2 we follow the same proof strategy as the onein [8]. However, it is not clear to us that a uniform bound on the multiplicityof edges can be treated with the same tools.The case of a d-regular bipartite graph G requires a more careful analysis,since the uniform probability space on the perfect matchings of G is not aproduct space as for complete graphs. We believe that it is possible to extendour result to sparser regular bipartite graphs, provided that every color doesnot appear too many times on the edges of the graph.Finally, the case of complete r–partite r–uniform hypergraphs is techni-cally more involved due to the higher degree of complexity of the hypergraphstructure, and all steps in the proof must be adapted to this setting.References[1] R. Aharoni, Ryser’s conjecture for tripartite 3-graphs. Combinatorica 21 (1)(2001) 1–4.[2] R. Aharoni and E. Berger, Rainbow matchings in r-partite graphs. ElectronicJ. Combinatorics, 2009.[3] R. Aharoni and P. Haxell, Hall’s theorem for hypergraphs. J. Graph Theory 35(2000), 83–88.[4] N. Alon, Multicolored matchings in hypergraphs, Moscow Journal ofCombinatorics and Number Theory 1 (2011) 3–10.[5] R. Bissacot, R. Ferna´ndez, A. Procacci and B. Scoppola, An improvement ofthe Lova´sz Local Lemma via cluster expansion. Combinatorics, Probability andComputing 20(5) (2011): 709–719.[6] A. E. Brouwer, A. J. de Vries and R. M. Wieringa, A lower bound for thelength of partial transversals in a Latin square. Nieuw Archief Voor Wiskunde,26 (1978), 330–332.[7] D. Clemens and J. Ehrenmu¨ller, An improved bound on the sizes of matchingsguaranteeing a rainbow matching. arXiv:1503.00438v1.[8] P. Erdo¨s and J. Spencer, Lopsided Lova´sz local lemma and Latin transversals.Discrete Appl. Math 30 (1991), no. 2-3, 151–154, ARIDAM III (New Brunswick,NJ, 1988).[9] R. Glebov, B. Sudakov and T. Szabo´, How many colors guarantee a Rainbowmatching?. Electronic J. Combinatorics 21 (1) (2014) 1–27.[10] D.G. Harris and A. Srinivasan. A constructive algorithm for the Lovsz LocalLemma on permutations. Proceedings of the Twenty-Fifth Annual ACM-SIAMSymposium on Discrete Algorithms. SIAM, 2014.[11] P. Hatami and P. W. Shor, A lower bound for the length of a partial transversalin a Latin square. J Combinatorics, Series A 115 (7) (2008) 1103–1113.[12] A. Kostochka and M. Yancey, Large rainbow matchings in edge-coloured graphs,Combinatorics, Probability and Computing 21 (2012), 255–263.[13] A. Lo, T.S. Tan, A note on large rainbow matchings in edge-coloured graphs,Graphs and Combinatorics 30, 2, (2014) 389–393.[14] G. Perarnau and O. Serra, Rainbow perfect matchings in complete bipartitegraphs: existence and counting. Combinatorics, Probability and Computing 22(5) (2013) 783–799.[15] H.J. Ryser, Neuere Probleme der Kombinatorik. Vortrogeber Kombinatorik,Oberwolfach, Matematisches Forschungsinstitute Obersolfach, 28–29 July 1967,pp. 69–91.[16] D. E. Woolbright, An n⇥n latin square has a transversal with at least n pndistinct symbols. J. Combinatorial Theory, Series A 24 (2) (1978) 235–237.

A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer Reviewe

Perarnau, Guillem

UPCommons

https://upcommons.upc.edu/bitstream/2117/101772/6/RainbowMatchingsJMDA.pdf

Rainbow perfect matchings in r-partite graph structures

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