On colouring random graphs

Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.

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On Komlós’ tiling theorem in random graphs

Combinatorics Probability Computing ◽

10.1017/s0963548319000129 ◽

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.

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The Minimum Vertex Degree of a Graph on Uniform Points in [0, 1]d

Advances in Applied Probability ◽

10.2307/1428077 ◽

1997 ◽

Vol 29 (3) ◽

pp. 582-594 ◽

Cited By ~ 23

Author(s):

Martin J. B. Appel ◽

Ralph P. Russo

Keyword(s):

Random Graph ◽

Nearest Neighbor ◽

Vertex Degree ◽

Degree Zero ◽

Asymptotic Results ◽

Limiting Behavior ◽

Edge Distance ◽

Vertex Set ◽

Image Position

This article continues an investigation begun in [2]. A random graph Gn(x) is constructed on independent random points U1, · ··, Un distributed uniformly on [0, 1]d, d ≧ 1, in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 < x < 1.Almost-sure asymptotic results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn(x) has no vertices of degree zero, is shown to satisfy Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.

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Metric Dimension for Random Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2639 ◽

2013 ◽

Vol 20 (4) ◽

Cited By ~ 5

Author(s):

Béla Bollobás ◽

Dieter Mitsche ◽

Paweł Prałat

Keyword(s):

Random Graph ◽

Random Graphs ◽

Metric Dimension ◽

Wide Range ◽

Minimum Number ◽

Vertex Set

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

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Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

Advances in Applied Probability ◽

10.1017/s0001867800000975 ◽

2006 ◽

Vol 38 (02) ◽

pp. 287-298 ◽

Cited By ~ 2

Author(s):

Maria Deijfen ◽

Ronald Meester

Keyword(s):

Probability Distribution ◽

Random Graph ◽

Random Graphs ◽

Random Number ◽

Random Variables ◽

Nearest Neighbors ◽

Expected Length ◽

Vertex Set ◽

Finitary Isomorphisms ◽

Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

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On the connectedness of a random graph

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100062034 ◽

1984 ◽

Vol 96 (1) ◽

pp. 151-166 ◽

Cited By ~ 12

Author(s):

G. R. Grimmett ◽

M. Keane ◽

J. M. Marstrand

Keyword(s):

Random Graph ◽

Random Graphs ◽

Higher Dimensions ◽

Greatest Common Divisor ◽

Vertex Set ◽

Vertex Sets

AbstractLet p = (p(i): i ≥ 0) be a sequence of numbers satisfying 0 ≤ p(i) < 1 for i = 0,1,2,…, and let G be a random graph with vertex set ℤ = {…, — 1, 0, 1,…} and with edge set defined as follows: for each pair i, j of vertices, where i ≤ j, there is an edge joining i and j with probability p(j — i), independently of the presence or absence of all other edges. We explore the connectedness of G, showing that G is almost surely connected if and only if Σip(i) = ∞ and the (positive) greatest common divisor of the set {i ≥ 1: p(i) < 0} equals 1; if one of these two conditions fails to hold then G is almost surely disconnected. Corresponding results hold in higher dimensions, for random graphs defined on the vertex sets ℤd where d ≥ 2.

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The Minimum Vertex Degree of a Graph on Uniform Points in [0, 1]d

Advances in Applied Probability ◽

10.1017/s0001867800028251 ◽

1997 ◽

Vol 29 (03) ◽

pp. 582-594

Author(s):

Martin J. B. Appel ◽

Ralph P. Russo

Keyword(s):

Random Graph ◽

Nearest Neighbor ◽

Vertex Degree ◽

Degree Zero ◽

Asymptotic Results ◽

Limiting Behavior ◽

Edge Distance ◽

Vertex Set ◽

Image Position

This article continues an investigation begun in [2]. A random graph Gn (x) is constructed on independent random points U 1, · ··, Un distributed uniformly on [0, 1] d , d ≧ 1, in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 < x < 1. Almost-sure asymptotic results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn (x) has no vertices of degree zero, is shown to satisfy Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.

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Constraining the Clustering Transition for Colorings of Sparse Random Graphs

The Electronic Journal of Combinatorics ◽

10.37236/7040 ◽

2018 ◽

Vol 25 (1) ◽

Cited By ~ 1

Author(s):

Michael Anastos ◽

Alan Frieze ◽

Wesley Pegden

Keyword(s):

Random Graph ◽

Random Graphs ◽

Hamming Distance ◽

Giant Component ◽

Vertex Set ◽

Almost All

Let $\Omega_q$ denote the set of proper $[q]$-colorings of the random graph $G_{n,m}, m=dn/2$ and let $H_q$ be the graph with vertex set $\Omega_q$ and an edge $\{\sigma,\tau\}$ where $\sigma,\tau$ are mappings $[n]\to[q]$ iff $h(\sigma,\tau)=1$. Here $h(\sigma,\tau)$ is the Hamming distance $|\{v\in [n]:\sigma(v)\neq\tau(v)\}|$. We show that w.h.p. $H_q$ contains a single giant component containing almost all colorings in $\Omega_q$ if $d$ is sufficiently large and $q\geq \frac{cd}{\log d}$ for a constant $c>3/2$.

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Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

Advances in Applied Probability ◽

10.1239/aap/1151337072 ◽

2006 ◽

Vol 38 (2) ◽

pp. 287-298 ◽

Cited By ~ 2

Author(s):

Maria Deijfen ◽

Ronald Meester

Keyword(s):

Probability Distribution ◽

Random Graph ◽

Random Graphs ◽

Random Number ◽

Random Variables ◽

Nearest Neighbors ◽

Expected Length ◽

Vertex Set ◽

Finitary Isomorphisms ◽

Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

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Random Graphs with Few Disjoint Cycles

Combinatorics Probability Computing ◽

10.1017/s0963548311000186 ◽

2011 ◽

Vol 20 (5) ◽

pp. 763-775 ◽

Cited By ~ 4

Author(s):

VALENTAS KURAUSKAS ◽

COLIN McDIARMID

Keyword(s):

Positive Integer ◽

Chromatic Number ◽

Random Graph ◽

Random Graphs ◽

Small Proportion ◽

Clique Number ◽

Disjoint Cycles ◽

Vertex Set ◽

Exponentially Small

The classical Erdős–Pósa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k + 1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G−B has no cycles. We show that, amongst all such graphs on vertex set {1,. . .,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class, concerning uniqueness of the blocker, connectivity, chromatic number and clique number.A key step in the proof of the main theorem is to show that there must be a blocker as in the Erdős–Pósa theorem with the extra ‘redundancy’ property that B–v is still a blocker for all but at most k vertices v ∈ B.

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