scholarly journals Edge-disjoint Hamilton cycles in random graphs

2013 ◽  
Vol 46 (3) ◽  
pp. 397-445 ◽  
Author(s):  
Fiachra Knox ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Random Graphs ◽  
Hamilton Cycles ◽  
Edge Disjoint

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at leastk

2000 ◽  
Vol 34 (1) ◽  
pp. 42-59 ◽  
Author(s):  
B. Bollob�s ◽  
C. Cooper ◽  
T. I. Fenner ◽  
A. M. Frieze
Keyword(s):  
Random Graphs ◽  
Minimum Degree ◽  
Hamilton Cycles ◽  
Edge Disjoint

scholarly journals An approximate version of Jackson’s conjecture

2020 ◽  
Vol 29 (6) ◽  
pp. 886-899
Author(s):  
Anita Liebenau ◽  
Yanitsa Pehova
Keyword(s):  
Bipartite Graph ◽  
Small Proportion ◽  
Complete Bipartite Graph ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Edge Disjoint ◽  
Bipartite Digraph ◽  
Bipartite Tournament ◽  
Approximate Version

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

Hamilton cycles in random graphs of minimal degree at least k

2012 ◽  
pp. 59-96 ◽  
Author(s):  
Béla Bollobás ◽  
T. I. Fenner ◽  
A. M. Frieze
Keyword(s):  
Random Graphs ◽  
Minimal Degree ◽  
Hamilton Cycles

scholarly journals Large triangle packings and Tuza’s conjecture in sparse random graphs

2020 ◽  
Vol 29 (5) ◽  
pp. 757-779 ◽  
Author(s):  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Shira Zerbib
Keyword(s):  
Greedy Algorithm ◽  
Random Graph ◽  
Random Graphs ◽  
Maximum Size ◽  
Packing Number ◽  
Edge Disjoint ◽  
Large Triangle

AbstractThe triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

scholarly journals Finding Hamilton cycles in random graphs with few queries

2016 ◽  
Vol 49 (4) ◽  
pp. 635-668 ◽  
Author(s):  
Asaf Ferber ◽  
Michael Krivelevich ◽  
Benny Sudakov ◽  
Pedro Vieira
Keyword(s):  
Random Graphs ◽  
Hamilton Cycles

scholarly journals Optimal covers with Hamilton cycles in random graphs

COMBINATORICA ◽  
2014 ◽  
Vol 34 (5) ◽  
pp. 573-596 ◽  
Author(s):  
Dan Hefetz ◽  
Daniela Kühn ◽  
John Lapinskas ◽  
Deryk Osthus
Keyword(s):  
Random Graphs ◽  
Hamilton Cycles

scholarly journals Packing Hamilton Cycles Online

2018 ◽  
Vol 27 (4) ◽  
pp. 475-495
Author(s):  
JOSEPH BRIGGS ◽  
ALAN FRIEZE ◽  
MICHAEL KRIVELEVICH ◽  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Hitting Time ◽  
Online Version ◽  
Hamilton Cycles ◽  
Fixed Integer ◽  
Edge Disjoint ◽  
Colour Class

It is known that w.h.p. the hitting time τ2σ for the random graph process to have minimum degree 2σ coincides with the hitting time for σ edge-disjoint Hamilton cycles [4, 9, 13]. In this paper we prove an online version of this property. We show that, for a fixed integer σ ⩾ 2, if random edges of Kn are presented one by one then w.h.p. it is possible to colour the edges online with σ colours so that at time τ2σ each colour class is Hamiltonian.

scholarly journals Edge Disjoint Hamilton Cycles in Knödel Graphs

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Palanivel Subramania Nadar Paulraja ◽  
S Sampath Kumar
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Cycle Decomposition ◽  
Edge Disjoint ◽  
International Audience ◽  
Appl Math

International audience The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

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