scholarly journals Hamilton Cycles in Random Graphs with a Fixed Degree Sequence

2010 ◽  
Vol 24 (2) ◽  
pp. 558-569 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Michael Krivelevich
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Hamilton Cycles ◽  
Fixed Degree

scholarly journals Karp–Sipser on Random Graphs with a Fixed Degree Sequence

2011 ◽  
Vol 20 (5) ◽  
pp. 721-741 ◽  
Author(s):  
TOM BOHMAN ◽  
ALAN FRIEZE
Keyword(s):  
Random Graphs ◽  
High Probability ◽  
Degree Sequence ◽  
Fixed Degree

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

SIR epidemics on random graphs with a fixed degree sequence

2012 ◽  
Vol 41 (2) ◽  
pp. 179-214 ◽  
Author(s):  
Tom Bohman ◽  
Michael Picollelli
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Fixed Degree

scholarly journals On the Chromatic Number of Random Graphs with a Fixed Degree Sequence

2007 ◽  
Vol 16 (05) ◽  
pp. 733 ◽  
Author(s):  
ALAN FRIEZE ◽  
MICHAEL KRIVELEVICH ◽  
CLIFF SMYTH
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Degree Sequence ◽  
Fixed Degree

scholarly journals Percolation on Random Graphs with a Fixed Degree Sequence

2022 ◽  
Vol 36 (1) ◽  
pp. 1-46
Author(s):  
Nikolaos Fountoulakis ◽  
Felix Joos ◽  
Guillem Perarnau
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Fixed Degree

scholarly journals Resilient Degree Sequences with respect to Hamilton Cycles and Matchings in Random Graphs

10.37236/8279 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Padraig Condon ◽  
Alberto Espuny Díaz ◽  
Daniela Kühn ◽  
Deryk Osthus ◽  
Jaehoon Kim
Keyword(s):  
Random Graphs ◽  
Perfect Matching ◽  
Degree Sequence ◽  
Hamilton Cycle ◽  
Vertex Degree ◽  
Degree Sequences ◽  
Hamilton Cycles ◽  
Degree Condition ◽  
Degree Conditions

Pósa's theorem states that any graph $G$ whose degree sequence $d_1 \le \cdots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e.~we prove a `resilience version' of Pósa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

scholarly journals The Critical Phase for Random Graphs with a Given Degree Sequence

2008 ◽  
Vol 17 (1) ◽  
pp. 67-86 ◽  
Author(s):  
M. KANG ◽  
T. G. SEIERSTAD
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Branching Process ◽  
Degree Sequence ◽  
Singularity Analysis ◽  
Fixed Degree ◽  
Critical Phase ◽  
Before And After

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence.

scholarly journals Random graphs with a given degree sequence

2011 ◽  
Vol 21 (4) ◽  
pp. 1400-1435 ◽  
Author(s):  
Sourav Chatterjee ◽  
Persi Diaconis ◽  
Allan Sly
Keyword(s):  
Random Graphs ◽  
Degree Sequence

scholarly journals How to Determine if a Random Graph with a Fixed Degree Sequence Has a Giant Component

2016 ◽  
Author(s):  
Felix Joos ◽  
Guillem Perarnau ◽  
Dieter Rautenbach ◽  
Bruce Reed
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Giant Component ◽  
Fixed Degree

First Occurrence of Hamilton Cycles in Random Graphs

1985 ◽  
pp. 173-178 ◽  
Author(s):  
M. Ajtai ◽  
J. Komlós ◽  
E. Szeraerédi
Keyword(s):  
Random Graphs ◽  
Hamilton Cycles ◽  
First Occurrence

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
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