Spanning structures and universality in sparse hypergraphs
In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Combin Probab Comput 9 (2000), 125–148] can be adapted from random graphs to random r‐uniform hypergraphs and provide sufficient conditions when a random r‐u...
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description | In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Combin Probab Comput 9 (2000), 125–148] can be adapted from random graphs to random r‐uniform hypergraphs and provide sufficient conditions when a random r‐uniform hypergraph H(r)(n,p) contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube‐hypergraphs, lattices, spheres, and Hamilton cycles in hypergraphs. Moreover, we study universality, i.e. when does an r‐uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by Δ? For H(r)(n,p) it is shown that this holds for p≫(lnn/n)1/Δ a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, Rödl and Ruciński [Random Struct Algorithms 46 (2015), 274–299] and of Ferber, Nenadov and Peter [Random Struct Algorithms 48 (2016), 546–564] and of Kim and Lee [SIAM J Discrete Math 28 (2014), 1467–1478]. Furthermore it is shown that the random graph G(n, p) for appropriate p and explicit constructions of universal graphs due to Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [Lecture Notes in Comput. Sci., Vol. 2129, Springer, Berlin, 2001, pp. 170–180] and Alon and Capalbo [Random Struct Algorithms 31 (2007), 123–133; Proceedings of the 9th Annual ACM‐SIAM Symposium Society of Industrial and Applied Mathematics, 2008, pp. 373–378] yield constructions of universal hypergraphs that are sparser than the random hypergraph H(r)(n,p) with p≫(lnn/n)1/Δ © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 819–844, 2016 |
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We notice that a general result of Riordan [Combin Probab Comput 9 (2000), 125–148] can be adapted from random graphs to random r‐uniform hypergraphs and provide sufficient conditions when a random r‐uniform hypergraph H(r)(n,p) contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube‐hypergraphs, lattices, spheres, and Hamilton cycles in hypergraphs. Moreover, we study universality, i.e. when does an r‐uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by Δ? For H(r)(n,p) it is shown that this holds for p≫(lnn/n)1/Δ a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, Rödl and Ruciński [Random Struct Algorithms 46 (2015), 274–299] and of Ferber, Nenadov and Peter [Random Struct Algorithms 48 (2016), 546–564] and of Kim and Lee [SIAM J Discrete Math 28 (2014), 1467–1478]. Furthermore it is shown that the random graph G(n, p) for appropriate p and explicit constructions of universal graphs due to Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [Lecture Notes in Comput. Sci., Vol. 2129, Springer, Berlin, 2001, pp. 170–180] and Alon and Capalbo [Random Struct Algorithms 31 (2007), 123–133; Proceedings of the 9th Annual ACM‐SIAM Symposium Society of Industrial and Applied Mathematics, 2008, pp. 373–378] yield constructions of universal hypergraphs that are sparser than the random hypergraph H(r)(n,p) with p≫(lnn/n)1/Δ © 2016 Wiley Periodicals, Inc. Random Struct. 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Alg</addtitle><description>In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Combin Probab Comput 9 (2000), 125–148] can be adapted from random graphs to random r‐uniform hypergraphs and provide sufficient conditions when a random r‐uniform hypergraph H(r)(n,p) contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube‐hypergraphs, lattices, spheres, and Hamilton cycles in hypergraphs. Moreover, we study universality, i.e. when does an r‐uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by Δ? For H(r)(n,p) it is shown that this holds for p≫(lnn/n)1/Δ a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, Rödl and Ruciński [Random Struct Algorithms 46 (2015), 274–299] and of Ferber, Nenadov and Peter [Random Struct Algorithms 48 (2016), 546–564] and of Kim and Lee [SIAM J Discrete Math 28 (2014), 1467–1478]. Furthermore it is shown that the random graph G(n, p) for appropriate p and explicit constructions of universal graphs due to Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [Lecture Notes in Comput. Sci., Vol. 2129, Springer, Berlin, 2001, pp. 170–180] and Alon and Capalbo [Random Struct Algorithms 31 (2007), 123–133; Proceedings of the 9th Annual ACM‐SIAM Symposium Society of Industrial and Applied Mathematics, 2008, pp. 373–378] yield constructions of universal hypergraphs that are sparser than the random hypergraph H(r)(n,p) with p≫(lnn/n)1/Δ © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 819–844, 2016</description><subject>Algorithms</subject><subject>Construction</subject><subject>Construction industry</subject><subject>Graphs</subject><subject>Lattices</subject><subject>Lectures</subject><subject>Mathematical models</subject><subject>random hypergraphs</subject><subject>spanning structures</subject><subject>thresholds</subject><subject>universality</subject><issn>1042-9832</issn><issn>1098-2418</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp10D1PwzAQBuAIgUQpDPyDSCwwpPVnbLNViBZQBYiCGC03cVqX1A12AuTf4xJgQGK6G573dHdRdAzBAAKAhs6rAQKpADtRDwLBE0Qg3932BCWCY7QfHXi_AgAwjHAvOp9VylpjF7GvXZPVjdM-VjaPG2vedJhWmrqNjY19pZzX8bKttFs4VS39YbRXqNLro-_aj57Gl48XV8n0bnJ9MZomGQEYJAxpjjNBCKVK0BwizjjRuhAUIggJnM9DxxDKNcpwITThKZ0TliEgckFAhvvRaTe3cpvXRvtaro3PdFkqqzeNl5BTilPKRBroyR-62jTOhu2CwpClHAMY1FmnMrfx3ulCVs6slWslBHL7RRnull9fDHbY2XdT6vZ_KB9mo59E0iWMr_XHb0K5F5kyzKh8vp1IdIPEZMrG8h5_AvKcgLo</recordid><startdate>201612</startdate><enddate>201612</enddate><creator>Parczyk, Olaf</creator><creator>Person, Yury</creator><general>Blackwell Publishing Ltd</general><general>Wiley Subscription Services, Inc</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>201612</creationdate><title>Spanning structures and universality in sparse hypergraphs</title><author>Parczyk, Olaf ; Person, Yury</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c4030-72e83c94455a95d128784eef95121141bb951722de2c3f9e4865b47c209d940c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Algorithms</topic><topic>Construction</topic><topic>Construction industry</topic><topic>Graphs</topic><topic>Lattices</topic><topic>Lectures</topic><topic>Mathematical models</topic><topic>random hypergraphs</topic><topic>spanning structures</topic><topic>thresholds</topic><topic>universality</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Parczyk, Olaf</creatorcontrib><creatorcontrib>Person, Yury</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Random structures & algorithms</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Parczyk, Olaf</au><au>Person, Yury</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Spanning structures and universality in sparse hypergraphs</atitle><jtitle>Random structures & algorithms</jtitle><addtitle>Random Struct. Alg</addtitle><date>2016-12</date><risdate>2016</risdate><volume>49</volume><issue>4</issue><spage>819</spage><epage>844</epage><pages>819-844</pages><issn>1042-9832</issn><eissn>1098-2418</eissn><abstract>In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Combin Probab Comput 9 (2000), 125–148] can be adapted from random graphs to random r‐uniform hypergraphs and provide sufficient conditions when a random r‐uniform hypergraph H(r)(n,p) contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube‐hypergraphs, lattices, spheres, and Hamilton cycles in hypergraphs. Moreover, we study universality, i.e. when does an r‐uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by Δ? For H(r)(n,p) it is shown that this holds for p≫(lnn/n)1/Δ a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, Rödl and Ruciński [Random Struct Algorithms 46 (2015), 274–299] and of Ferber, Nenadov and Peter [Random Struct Algorithms 48 (2016), 546–564] and of Kim and Lee [SIAM J Discrete Math 28 (2014), 1467–1478]. Furthermore it is shown that the random graph G(n, p) for appropriate p and explicit constructions of universal graphs due to Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [Lecture Notes in Comput. Sci., Vol. 2129, Springer, Berlin, 2001, pp. 170–180] and Alon and Capalbo [Random Struct Algorithms 31 (2007), 123–133; Proceedings of the 9th Annual ACM‐SIAM Symposium Society of Industrial and Applied Mathematics, 2008, pp. 373–378] yield constructions of universal hypergraphs that are sparser than the random hypergraph H(r)(n,p) with p≫(lnn/n)1/Δ © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 819–844, 2016</abstract><cop>Hoboken</cop><pub>Blackwell Publishing Ltd</pub><doi>10.1002/rsa.20690</doi><tpages>26</tpages><oa>free_for_read</oa></addata></record> |
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subjects | Algorithms Construction Construction industry Graphs Lattices Lectures Mathematical models random hypergraphs spanning structures thresholds universality |
title | Spanning structures and universality in sparse hypergraphs |
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