scholarly journals On Resilience of Connectivity in the Evolution of Random Graphs

10.37236/7885 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Luc Haller ◽  
Miloš Trujić
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Hitting Time ◽  
Connected Component ◽  
Empty Graph ◽  
Spanning Subgraph ◽  
Time Result ◽  
Graph Property

In this note we establish a resilience version of the classical hitting time result of Bollobás and Thomason regarding connectivity. A graph $G$ is said to be $\alpha$-resilient with respect to a monotone increasing graph property $\mathcal{P}$ if for every spanning subgraph $H \subseteq G$ satisfying $\deg_H(v) \leqslant \alpha \deg_G(v)$ for all $v \in V(G)$, the graph $G - H$ still possesses $\mathcal{P}$. Let $\{G_i\}$ be the random graph process, that is a process where, starting with an empty graph on $n$ vertices $G_0$, in each step $i \geqslant 1$ an edge $e$ is chosen uniformly at random among the missing ones and added to the graph $G_{i - 1}$. We show that the random graph process is almost surely such that starting from $m \geqslant (\tfrac{1}{6} + o(1)) n \log n$, the largest connected component of $G_m$ is $(\tfrac{1}{2} - o(1))$-resilient with respect to connectivity. The result is optimal in the sense that the constants $1/6$ in the number of edges and $1/2$ in the resilience cannot be improved upon. We obtain similar results for $k$-connectivity.

scholarly journals Client-Waiter Games on Complete and Random Graphs

10.37236/6039 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Oren Dean ◽  
Michael Krivelevich
Keyword(s):  
Positive Integer ◽  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Upper Bounds ◽  
Connected Component ◽  
Lower And Upper Bounds ◽  
Large Connected Component ◽  
Graph Property ◽  
Player Game

For a graph $ G $, a monotone increasing graph property $ \mathcal{P} $ and positive integer $ q $, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most $ q+1 $ unclaimed edges of $ G $ from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property $ \mathcal{P} $ by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the $ H $-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the $ H $-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

The Largest Component in Subcritical Inhomogeneous Random Graphs

2010 ◽  
Vol 20 (1) ◽  
pp. 131-154 ◽  
Author(s):  
TATYANA S. TUROVA
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Graph Model ◽  
Exact Formula ◽  
Connected Component ◽  
Subcritical Case ◽  
Random Graph Model ◽  
Asymptotic Size ◽  
Inhomogeneous Random Graphs ◽  
Inhomogeneous Random Graph

We study the ‘rank 1 case’ of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result complements the corresponding known result in the supercritical case. We provide some examples of applications of the derived formula.

scholarly journals A spanning bandwidth theorem in random graphs

2021 ◽  
pp. 1-31
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Julia Ehrenmüller ◽  
Jakob Schnitzer ◽  
Anusch Taraz
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Spanning Subgraph ◽  
Vertex Graph ◽  
Special Case

Abstract The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

scholarly journals Triangle-Free Subgraphs of Random Graphs

2017 ◽  
Vol 27 (2) ◽  
pp. 141-161
Author(s):  
PETER ALLEN ◽  
JULIA BÖTTCHER ◽  
YOSHIHARU KOHAYAKAWA ◽  
BARNABY ROBERTS
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Random Graphs ◽  
Minimum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Spanning Subgraph

Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p−1n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + ϵ)pn is O(p−1n)-close to r-partite for some r = r(ϵ). These are random graph analogues of a result by Andrásfai, Erdős and Sós (Discrete Math.8 (1974), 205–218), and a result by Thomassen (Combinatorica22 (2002), 591–596). We also show that our results are best possible up to a constant factor.

scholarly journals Maximizing the Size of the Giant

2012 ◽  
Vol 49 (4) ◽  
pp. 1156-1165 ◽  
Author(s):  
Tom Britton ◽  
Pieter Trapman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Related Question ◽  
Configuration Model ◽  
Connected Component ◽  
The Social ◽  
Random Weight ◽  
The Mean ◽  
Positive Integers ◽  
Outbreak Size

Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring ‘on’ the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight Xi with E(Xi)=µ, and where there is an edge between i and j with probability 1-e-XiXj/(µ n), independently of other edges. The second model is the thinned configuration model in which the n vertices of the ground graph have i.i.d. ground degrees, distributed as D, with E(D) = µ. The graph of interest is obtained by deleting edges independently with probability 1-p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1-q, where q depends on X or D and p. We investigate for which distributions X and D with given µ and p, 1-q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.

Maximizing the Size of the Giant

2012 ◽  
Vol 49 (04) ◽  
pp. 1156-1165
Author(s):  
Tom Britton ◽  
Pieter Trapman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Related Question ◽  
Configuration Model ◽  
Connected Component ◽  
The Social ◽  
Random Weight ◽  
The Mean ◽  
Positive Integers ◽  
Outbreak Size

Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring ‘on’ the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight X i with E(X i )=µ, and where there is an edge between i and j with probability 1-e-X i X j /(µ n), independently of other edges. The second model is the thinned configuration model in which the n vertices of the ground graph have i.i.d. ground degrees, distributed as D, with E(D) = µ. The graph of interest is obtained by deleting edges independently with probability 1-p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1-q, where q depends on X or D and p. We investigate for which distributions X and D with given µ and p, 1-q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.

Spanning Subgraphs of Random Graphs

2000 ◽  
Vol 9 (2) ◽  
pp. 125-148 ◽  
Author(s):  
OLIVER RIORDAN
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Moment Method ◽  
Subgraph Isomorphic ◽  
Square Grid ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Fixed Probability ◽  
Second Moment Method ◽  
The Second Moment Method

Let Gp be a random graph on 2d vertices where edges are selected independently with a fixed probability p > ¼, and let H be the d-dimensional hypercube Qd. We answer a question of Bollobás by showing that, as d → ∞, Gp almost surely has a spanning subgraph isomorphic to H. In fact we prove a stronger result which implies that the number of d-cubes in G ∈ [Gscr ](n, M) is asymptotically normally distributed for M in a certain range. The result proved can be applied to many other graphs, also improving previous results for the lattice, that is, the 2-dimensional square grid. The proof uses the second moment method – writing X for the number of subgraphs of G isomorphic to H, where G is a suitable random graph, we expand the variance of X as a sum over all subgraphs of H itself. As the subgraphs of H may be quite complicated, most of the work is in estimating the various terms of this sum.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 271-300 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Lower Bound ◽  
Random Graphs ◽  
Empty Graph ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove a lower bound ofnβ(F,r)on the typical duration of this game, where β(F,r) is a function that is strictly increasing inrand satisfies limr→∞β(F,r) = 2 − 1/m2(F), wheren2−1/m2(F)is the threshold of the corresponding offline colouring problem.

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