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Hamilton Cycles In Line Graphs Of 3-Hypergraphs

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We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry -quasirandom 3-graphs. Our first result 1 Introduction Hamilton cycles occupy a position of central importance in graph theory, and are the subject of countless results. The most famous is of course Dirac’s Theorem [6], which states that a Hamilton cycle can always be found in any n-vertex graph with all degrees at least n=2. Much more work has been done to determine conditions for Hamiltonicity in graphs, digraphs,

In 1931, Whitney [11] proved that every 4-connected triangulation of the plane contains a Hamilton cycle. In 1956, Tutte [10] proved a more general result: every 4-connected planar graph contains a Hamilton cycle. However, 3-connected planar graphs need not contain Hamilton cycles. For such examples, see Holton and McKay [7].

Ore-type condition for Hamilton ℓ-cycle in k-uniform hypergraphs

(PDF) Closing Gaps in Problems related to Hamilton Cycles in Random ...

It is proved that Thomassen’s conjecture (every 4-connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4- connection is 1- Hamilton-connected and/or 2-edge-Hamilton-connected.

They proved that every 5-connected line graph of minimum degree at least 6 is Hamilton-connected in [3], and every 3-connected essentially 9-connected line graph is Hamilton-connected in [4]. Hamilton cycles in line graphs of 3-hypergraphs Abstract We prove that every 52-connected line graph of a rank 3 hypergraph is Hamiltonian. This is the first result of this type for hypergraphs of bounded rank other than ordinary graphs. Many theorems and concepts involving graphs also hold for hypergraphs, in particular: Matching in hypergraphs; Vertex cover in hypergraphs (also known as: transversal); Line graph of a hypergraph; Hypergraph grammar – created by augmenting a class of hypergraphs with a set of replacement rules; Ramsey’s theorem; Erdős–Ko–Rado theorem; Kruskal–Katona theorem on

We develop a framework to study minimum d-degree conditions in k-uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path-cover and connecting arguments for all k and d at once, Hamilton cycles in graphs and hypergraphs: Hamilton cycle contains every vertex exactly once related to the Traveling salesman problem Recommendations Hourglasses and Hamilton cycles in 4-connected claw-free graphs Hamilton cycles in line graphs of 3-hypergraphs Discrepancies of spanning trees and Hamilton cycles Comments

  • Random graphs and hypergraphs for complex networks
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  • Positive codegree thresholds for Hamilton cycles in hypergraphs

Utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Király, and Kriesell in 2003, we generalize the above We study sufficient conditions for the existence of Hamilton cycles in uniformly dense 3-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles, and Aigner-Horev and Levy considered them for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal Request PDF | On hamiltonian line graphs of hypergraphs | A graph is supereulerian if it has a spanning eulerian subgraph. Harary and Nash-Williams in 1968 proved that the line graph of a graph G

Abstract It is well known that if a graph G contains a spanning closed trail, then its line graph L (G) is Hamiltonian. In this note, it is proved that if a graph G with minimum degree at least 4k has k edge-disjoint spanning closed trails, then AbstractWe prove that for every ε > 0 there exists n 0 = n 0 ( ε ) such that every regular oriented graph on n > n 0 vertices and degree at least ( 1 / 4 + ε ) n has a Hamilton cycle. This establis As is the case with graphs, there is a large body of research studying various Dirac-type conditions in hypergraphs, that is smallest -degree which implies existence of Hamilton -cycles (e.g., [46, 60, 61], surveys [49, 64] and the references within).

Every 4-connected line-graph i s hamiltonian ( i f G is a cubic cyclically-4-edge connected graph, its line-graph L(G) is 4-connected, and a Hamilton cycle of L(G) easily yields a dominating cycle of G). We prove that for every ε > 0 there exists n 0 = n 0 ( ε ) such that every regular oriented graph on n > n 0 vertices and degree at least ( 1 / 4 + ε ) n has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed The study of Hamilton cycles is an important topic in graph theory. In recent years, researchers have worked on extending the classical theorem of Dirac [7] on Hamilton cycles to hypergraphs – see recent surveys of [23], [20].

Saito AXiong L (2009)Closure, stability and iterated line graphs with a 2-factorDiscrete Mathematics 10.1016/j.disc.2009.03.003309:16 (5000-5010)Online publication date: 1-Aug-2009

(PDF) Efficient solution for finding Hamilton cycles in undirected graphs

Hamilton Cycles in the Line Graph of a Random Hypergraph Katarzyna Rybarczyk P2.17 PDF

Recommendations Spanning subgraphs in graphs and hypergraphs Hamilton cycles in line graphs of 3-hypergraphs Abstract We prove that every 52-connected line graph of a rank 3 hypergraph is Hamiltonian. This is the first result of this type for hypergraphs of bounded rank other than ordinary graphs.

A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac’s theorem in mind, it is natural to ask what minimum $$ d $$ -degree condition guarantees the existence of a loose Hamilton cycle in a $$ k $$ -uniform hypergraph. For $$ k=3 $$ and each A lot of resilience results are known for random graphs. For instance, the containment of triangle factors [4], almost spanning trees of bounded degree [3], pancyclic graphs [15], almost spanning and spanning bounded degree graphs with sublinear bandwidth [1, 6, 13], directed Hamilton cycles [11, 12, 18], perfect matchings and Hamilton cycles in random graph processes [19, 20],

A fundamental theorem of Dirac [3] states that any graph on vertices with minimum degree at least contains a Hamilton cycle. A natural question is whether this theorem can be extended to hypergraphs. This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains some material on closely related topics such as traceable, pancyclic and Hamiltonian connected graphs. We study sufficient conditions for the existence of Hamilton cycles in uniformly dense 3-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles, and Aigner-Horev and Levy considered them

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness. Bollobás [3] and Bondy [4] asked for an asymptotic estimate for the number of dis-tinct Hamilton cycles in Dirac graphs. In 2003, Sárközy, Selkow, and Szemerédi [29] made substantial progress on this question by showing that n-vertex Dirac graphs con-tain exp(n ln n (n)) Hamilton cycles. They also posed the question of whether this is the right order of magnitude for graphs

DECOMPOSITION INTO CYCLES 1: HAMILTON DECOMPOSITIONS B. ALSPACH Department of Mathematics and Statistics Simon Fraser University Burnaby, B.C., V5A 1S6 Canada J.-C. BERMOND; D. SOTTEAU C.N.R.S., UA410 Informatique mt. 490 Université Paris-Sud 91405 Orsay France Abstract In this part we survey the results concerning the partitions of the edge A Hamilton cycle decomposition of a graph G = ( V, E ) is a partition of E into Hamilton cycles. Let K r × n k denote the complete n-balanced r-partite k -uniform hypergraph.