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Hamilton cycles in sparse robustly expanding digraphs [article]

Allan Lo, Viresh Patel
2018 arXiv   pre-print
We give a proof that does not use the Regularity Lemma and, indeed, we can apply our result to suitable sparse robustly expanding digraphs.  ...  These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle.  ...  In Section 5 we show that the vertices of any robustly expanding digraph can be covered by a small number of cycles.  ... 
arXiv:1507.04472v2 fatcat:42a5hsxowregbiaiti3zwforx4

Hamilton Cycles in Sparse Robustly Expanding Digraphs

Allan Lo, Viresh Patel
2018 Electronic Journal of Combinatorics  
These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle.  ...  The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs.  ...  In Section 5 we show that the vertices of any robustly expanding digraph can be covered by a small number of cycles.  ... 
doi:10.37236/7418 fatcat:wbcyplbmcfdi7jobrj25xnucre

Approximate Hamilton decompositions of robustly expanding regular digraphs [article]

Deryk Osthus, Katherine Staden
2013 arXiv   pre-print
G contains a set of r-o(r) edge-disjoint Hamilton cycles.  ...  We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e.  ...  It will also be convenient to use the following result from [15] , which guarantees a Hamilton cycle in a robustly expanding digraph.  ... 
arXiv:1206.2810v2 fatcat:kpfpxwfuubc2fcw5zpv5h7m7sy

Approximate Hamilton Decompositions of Robustly Expanding Regular Digraphs

Deryk Osthus, Katherine Staden
2013 SIAM Journal on Discrete Mathematics  
G contains a set of r − o(r) edge-disjoint Hamilton cycles.  ...  We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e.  ...  It will also be convenient to use the following result from [15] , which guarantees a Hamilton cycle in a robustly expanding digraph.  ... 
doi:10.1137/120880951 fatcat:sw2x65wog5ce3n5t2iuwb3ziau

Hamilton decompositions of regular expanders: Applications

Daniela Kühn, Deryk Osthus
2014 Journal of combinatorial theory. Series B (Print)  
In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton  ...  ) a proof of a conjecture of Erdos on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.  ...  Robust expanders. A graph or digraph G has a Hamilton decomposition if it contains a set of edge-disjoint Hamilton cycles which together cover all the edges of G.  ... 
doi:10.1016/j.jctb.2013.10.006 fatcat:6jr47nkcs5fc3gciuqemj6qkne

Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

DANIELA KÜHN, JOHN LAPINSKAS, DERYK OSTHUS
2012 Combinatorics, probability & computing  
Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.  ...  We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ = (1/2+α)n.  ...  Our first aim is to find a sparse even factor H of G which is a robust expander. This has essentially already been done in [12] , but for digraphs.  ... 
doi:10.1017/s0963548312000569 fatcat:74cvhk3lq5fznechirr24pzlje

Optimal packings of Hamilton cycles in graphs of high minimum degree [article]

Daniela Kühn, John Lapinskas, Deryk Osthus
2012 arXiv   pre-print
Our proof relies on a recent and very general result of K\"uhn and Osthus on Hamilton decomposition of robustly expanding regular graphs.  ...  We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n.  ...  Our first aim is to find a sparse even factor H of G which is a robust expander. This has essentially already been done in [12] , but for digraphs.  ... 
arXiv:1211.3263v1 fatcat:vogmrskeorhvphb2p47ss7gpei

Hamilton decompositions of regular bipartite tournaments [article]

Bertille Granet
2022 arXiv   pre-print
In 1981, Jackson conjectured that any regular bipartite tournament can be decomposed into Hamilton cycles. We prove this conjecture for all sufficiently large bipartite tournaments.  ...  Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.  ...  In particular, the author is very grateful to Allan Lo for sharing ideas and to Daniela Kühn and Deryk Osthus for supplying guidance and feedback.  ... 
arXiv:2209.02988v1 fatcat:yk6xrj24knfd3axkr3r7immf3q

Hamilton decompositions of regular expanders: A proof of Kelly's conjecture for large tournaments

Daniela Kühn, Deryk Osthus
2013 Advances in Mathematics  
We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles.  ...  This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdős on packing Hamilton cycles in random tournaments.  ...  Acknowledgements We would like to thank John Lapinskas for an idea which led to a simplification of the cycle absorbing argument.  ... 
doi:10.1016/j.aim.2013.01.005 fatcat:abtzn4ovljgwxjowcoilviv23y

Hamilton cycles in graphs and hypergraphs: an extremal perspective [article]

Daniela Kühn, Deryk Osthus
2014 arXiv   pre-print
New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs.  ...  As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research.  ...  This notion was introduced (for digraphs) by Kühn, Osthus and Treglown [95] , who showed that every robustly expanding digraph of linear minimum in-and outdegree contains a Hamilton cycle.  ... 
arXiv:1402.4268v3 fatcat:fuwonzdcqrc7xinisddgengmqq

Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments [article]

Daniela Kühn, Deryk Osthus
2013 arXiv   pre-print
We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles.  ...  This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdos on packing Hamilton cycles in random tournaments.  ...  Similarly, in Section 5 we collect general properties of robustly expanding digraphs. Section 6 is devoted to tools for finding Hamilton cycles (in robustly expanding digraphs).  ... 
arXiv:1202.6219v2 fatcat:3fp7frz4kvg4rjueelxm3e25uy

Proof of the 1-factorization and Hamilton Decomposition Conjectures [article]

Béla Csaba, Daniela Kühn, Allan Lo, Deryk Osthus, Andrew Treglown
2014 arXiv   pre-print
(iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ> n/2. Then G contains at least reg_ even(n,δ)/2 > (n-2)/8 edge-disjoint Hamilton cycles.  ...  (ii) [Hamilton decomposition conjecture] Suppose that D > n/2 . Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching.  ...  Recall that this guarantees the existence of a 'robustly decomposable' digraph G rob dir , whose crucial property is that H + G rob dir has a Hamilton decomposition for any sparse regular digraph H which  ... 
arXiv:1401.4159v2 fatcat:ghtaawuykjgnjfo6cxso6tyl3e

Proof of the 1-factorization and Hamilton decomposition conjectures [chapter]

Béla Csaba, Daniela Kühn, Allan Lo, Deryk Osthus, Andrew Treglown
2013 The Seventh European Conference on Combinatorics, Graph Theory and Applications  
Recall that this guarantees the existence of a 'robustly decomposable' digraph G rob dir , whose crucial property is that H + G rob dir has a Hamilton decomposition for any sparse regular digraph H which  ...  As before, the lemma guarantees the existence of a 'robustly decomposable' digraph G rob dir , whose crucial property is that H + G rob dir has a Hamilton decomposition for any sparse bipartite regular  ...  Merging Cycles to Obtain Hamilton Cycles. Recall that we have removed a sparse subdigraph H from G and that G ′ = G − H.  ... 
doi:10.1007/978-88-7642-475-5_76 fatcat:nd463436gnbszfw4z2pgxq2q7a

Proof of the 1-factorization and Hamilton Decomposition Conjectures

Béla Csaba, Daniela Kühn, Allan Lo, Deryk Osthus, Andrew Treglown
2016 Memoirs of the American Mathematical Society  
Recall that this guarantees the existence of a 'robustly decomposable' digraph G rob dir , whose crucial property is that H + G rob dir has a Hamilton decomposition for any sparse regular digraph H which  ...  As before, the lemma guarantees the existence of a 'robustly decomposable' digraph G rob dir , whose crucial property is that H + G rob dir has a Hamilton decomposition for any sparse bipartite regular  ...  Merging Cycles to Obtain Hamilton Cycles. Recall that we have removed a sparse subdigraph H from G and that G = G − H.  ... 
doi:10.1090/memo/1154 fatcat:evla77oqnnbizk2yrc5odoxymm

Optimal covers with Hamilton cycles in random graphs [article]

Dan Hefetz and Daniela Kühn and John Lapinskas and Deryk Osthus
2013 arXiv   pre-print
A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G.  ...  Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.  ...  The result in [15] is based on a recent result of Kühn and Osthus [14] which guarantees the existence of a Hamilton decomposition in every regular 'robustly expanding' digraph.  ... 
arXiv:1203.3868v2 fatcat:5yqlvgzcdfgttlclt57qe7i6dm
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