scholarly journals Cover time of a random graph with given degree sequence

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Mohammed Abdullah ◽  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Average Degree ◽  
Cover Time ◽  
Vertex Set ◽  
International Audience

International audience In this paper we establish the cover time of a random graph $G(\textbf{d})$ chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\textbf{d}$. We show that under certain restrictions on $\textbf{d}$, the cover time of $G(\textbf{d})$ is with high probability asymptotic to $\frac{d-1}{ d-2} \frac{\theta}{ d}n \log n$. Here $\theta$ is the average degree and $d$ is the $\textit{effective minimum degree}$. The effective minimum degree is the first entry in the sorted degree sequence which occurs order $n$ times.

scholarly journals Blocks in Constrained Random Graphs with Fixed Average Degree

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Average Degree ◽  
Additional Restriction ◽  
Structural Constraints ◽  
Analytic Framework ◽  
Discrete Algorithms ◽  
International Audience ◽  
Sharp Concentration

International audience This work is devoted to the study of typical properties of random graphs from classes with structural constraints, like for example planar graphs, with the additional restriction that the average degree is fixed. More precisely, within a general analytic framework, we provide sharp concentration results for the number of blocks (maximal biconnected subgraphs) in a random graph from the class in question. Among other results, we discover that essentially such a random graph belongs with high probability to only one of two possible types: it either has blocks of at most logarithmic size, or there is a \emphgiant block that contains linearly many vertices, and all other blocks are significantly smaller. This extends and generalizes the results in the previous work [K. Panagiotou and A. Steger. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432-440, 2009], where similar statements were shown without the restriction on the average degree.

scholarly journals Upper bounds on the non-3-colourability threshold of random graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Upper Bounds ◽  
Average Degree ◽  
Expected Number ◽  
International Audience ◽  
Full Analysis ◽  
A Minor ◽  
Minor Improvement

International audience We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.

scholarly journals On the Strength of Connectedness of a Random Hypergraph

10.37236/4666 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Daniel Poole
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Stopping Time ◽  
Minimal Degree ◽  
Vertex Set ◽  
Random Hypergraph

Bollobás and Thomason (1985) proved that for each $k=k(n) \in [1, n-1]$, with high probability, the random graph process, where edges are added to vertex set $V=[n]$ uniformly at random one after another, is such that the stopping time of having minimal degree $k$ is equal to the stopping time of becoming $k$-(vertex-)connected. We extend this result to the $d$-uniform random hypergraph process, where $k$ and $d$ are fixed. Consequently, for $m=\frac{n}{d}(\ln n +(k-1)\ln \ln n +c)$ and $p=(d-1)! \frac{\ln n + (k-1) \ln \ln n +c}{n^{d-1}}$, the probability that the random hypergraph models $H_d(n, m)$ and $H_d(n, p)$ are $k$-connected tends to $e^{-e^{-c}/(k-1)!}.$

scholarly journals Long Paths and Cycles in Random Subgraphs of $\mathcal{H}$-Free Graphs

10.37236/3198 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Wojciech Samotij
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Minimum Degree ◽  
Average Degree ◽  
Free Graph ◽  
Connected Graphs ◽  
Random Subgraph ◽  
Minimum Number ◽  
Random Subgraphs ◽  
Long Paths

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

scholarly journals On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Shih-Yan Chen ◽  
Shin-Shin Kao ◽  
Hsun Su
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Sequence Characterization ◽  
Hamiltonian Connected ◽  
Vertex Graph ◽  
Minimum Number ◽  
Hamiltonian Connectivity ◽  
International Audience ◽  
Delta G ◽  
Hamiltonian Properties

International audience Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called <i>$k$-vertex fault traceable</i>, <i>$k$-vertex fault Hamiltonian</i>, or <i>$k$-vertex fault Hamiltonian-connected</i> if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.

scholarly journals Cover time of a random graph with a degree sequence II: Allowing vertices of degree two

2014 ◽  
Vol 45 (4) ◽  
pp. 627-674
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Eyal Lubetzky
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Cover Time

scholarly journals Cover time of a random graph with given degree sequence

2012 ◽  
Vol 312 (21) ◽  
pp. 3146-3163 ◽  
Author(s):  
Mohammed Abdullah ◽  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Cover Time

scholarly journals Subgraphs of Dense Random Graphs with Specified Degrees

2011 ◽  
Vol 20 (3) ◽  
pp. 413-433 ◽  
Author(s):  
BRENDAN D. McKAY
Keyword(s):  
Saddle Point ◽  
Random Graph ◽  
Random Graphs ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Average Degree ◽  
Point Method ◽  
Saddle Point Method ◽  
Induced Subgraph ◽  
Basic Facts

Let d = (d1, d2, . . ., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n.Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).

scholarly journals Counting Trees in Graphs

10.37236/6084 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Jacques Verstraete ◽  
Dhruv Mubayi
Keyword(s):  
Lower Bound ◽  
Convex Function ◽  
Disjoint Union ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Average Degree ◽  
Vertex Graph ◽  
Counting Trees

Erdős and Simonovits proved that the number of paths of length $t$ in an $n$-vertex graph of average degree $d$ is at least $(1 - \delta) nd(d - 1) \cdots (d - t + 1)$, where $\delta = (\log d)^{-1/2 + o(1)}$ as $d \rightarrow \infty$. In this paper, we strengthen and generalize this result as follows. Let $T$ be a tree with $t$ edges. We prove that for any $n$-vertex graph $G$ of average degree $d$ and minimum degree greater than $t$, the number of labelled copies of $T$ in $G$ is at least \[(1 - \varepsilon) n d(d - 1) \cdots (d - t + 1)\] where $\varepsilon = O(d^{-2})$ as $d \rightarrow \infty$. This bound is tight except for the term $1 - \varepsilon$, as shown by a disjoint union of cliques. Our proof is obtained by first showing a lower bound that is a convex function of the degree sequence of $G$, and this answers a question of Dellamonica et. al.

scholarly journals On the Largest Component of a Hyperbolic Model of Complex Networks

10.37236/4958 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Michel Bode ◽  
Nikolaos Fountoulakis ◽  
Tobias Müller
Keyword(s):  
Complex Networks ◽  
Uniform Distribution ◽  
Random Graph ◽  
Power Law ◽  
High Probability ◽  
Degree Sequence ◽  
Hyperbolic Model ◽  
Hyperbolic Distance ◽  
Component Structure ◽  
Two Parameters

We consider a model for complex networks that was introduced by Krioukov et al.  In this model, $N$ points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an  edge if they are within a certain hyperbolic distance.  The $N$ points are distributed according to a quasi-uniform distribution, which is a distorted version of  the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering. The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power-law and $\nu$ controls the average degree. The present paper focuses on the evolution of the component structure of the random graph.  We show that (a) for $\alpha > 1$ and $\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for $\alpha < 1$ and $\nu$ arbitrary with high probability there is a "giant" component  of linear order,  and (c) when $\alpha=1$ then there is a non-trivial phase transition for the existence of a linear-sized component in terms of $\nu$. A corrigendum was added to this paper 29 Dec 2018.

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