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 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.

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.

Threshold Functions for H-factors

1993 ◽  
Vol 2 (2) ◽  
pp. 137-144 ◽  
Author(s):  
Noga Alon ◽  
Raphael Yuster
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Threshold Function ◽  
Sharp Threshold ◽  
Threshold Functions ◽  
Spanning Subgraph ◽  
Degree Of A Vertex ◽  
Vertex Disjoint ◽  
Special Case

Let H be a graph on h vertices, and G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractional arboricity of H is , where the maximum is taken over all subgraphs (V′, E′) of H with |V′| > 1. Let δ(H) denote the minimum degree of a vertex of H. It is shown that if δ(H) < a(H), then n−1/a(H) is a sharp threshold function for the property that the random graph G(n, p) contains an H-factor. That is, there are two positive constants c and C so that for p(n) = cn−1/a(H) almost surely G(n, p(n)) does not have an H-factor, whereas for p(n) = Cn−1/a(H), almost surely G(n, p(n)) contains an H-factor (provided h divides n). A special case of this answers a problem of Erdős.

scholarly journals Expanders Are Universal for the Class of All Spanning Trees

2013 ◽  
Vol 22 (2) ◽  
pp. 253-281 ◽  
Author(s):  
DANIEL JOHANNSEN ◽  
MICHAEL KRIVELEVICH ◽  
WOJCIECH SAMOTIJ
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Clique Number ◽  
Interesting Consequence ◽  
Random Regular Graph ◽  
Vertex Graph ◽  
Graph Class ◽  
Universal Graphs ◽  
Natural Expansion

A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

scholarly journals The bandwidth theorem for locally dense graphs

2020 ◽  
Vol 8 ◽  
Author(s):  
Katherine Staden ◽  
Andrew Treglown
Keyword(s):  
Maximum Degree ◽  
Blow Up ◽  
Minimum Degree ◽  
Bounded Degree ◽  
Dense Graphs ◽  
Vertex Graph ◽  
Delta G

Abstract The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás andKomlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. In this paper, we prove a version of the bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense n-vertex graph G with $\delta (G)> (1/2+o(1))n$ contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.

scholarly journals A Note on the Acquaintance Time of Random Graphs

10.37236/3357 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
William B. Kinnersley ◽  
Dieter Mitsche ◽  
Paweł Prałat
Keyword(s):  
Lower Bound ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Short Note ◽  
Vertex Graph ◽  
Small Improvement

In this short note, we prove the conjecture of Benjamini, Shinkar, and Tsur on the acquaintance time $\mathcal{AC}(G)$ of a random graph $G \in G(n,p)$. It is shown that asymptotically almost surely $\mathcal{AC}(G) = O(\log n / p)$ for $G \in G(n,p)$, provided that $pn > (1+\epsilon) \log n$ for some $\epsilon > 0$ (slightly above the threshold for connectivity). Moreover, we show a matching lower bound for dense random graphs, which also implies that asymptotically almost surely $K_n$ cannot be covered with $o(\log n / p)$ copies of a random graph $G \in G(n,p)$, provided that $pn > n^{1/2+\epsilon}$ and $p < 1-\epsilon$ for some $\epsilon>0$. We conclude the paper with a small improvement on the general upper bound showing that for any $n$-vertex graph $G$, we have $\mathcal{AC}(G) = O(n^2/\log n )$.

scholarly journals On the Strong Chromatic Number of Random Graphs

2008 ◽  
Vol 17 (2) ◽  
pp. 271-286 ◽  
Author(s):  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Maximum Degree ◽  
Disjoint Sets ◽  
Strong Chromatic Number ◽  
Proper Vertex

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colourable if, for every partition of V(G) into disjoint sets V1 ∪ ··· ∪ Vr, all of size exactly k, there exists a proper vertex k-colouring of G with each colour appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colourable if the graph obtained by adding $k \big\lceil \frac{n}{k} \big\rceil - n$ isolated vertices is strongly k-colourable. The strong chromatic number of G is the minimum k for which G is strongly k-colourable. In this paper, we study the behaviour of this parameter for the random graph Gn,p. In the dense case when p ≫ n−1/3, we prove that the strong chromatic number is a.s. concentrated on one value Δ + 1, where Δ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

scholarly journals Logconcave Random Graphs

10.37236/380 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Alan Frieze ◽  
Santosh Vempala ◽  
Juan Vera
Keyword(s):  
Combinatorial Optimization ◽  
Random Graph ◽  
Random Graphs ◽  
Optimization Problem ◽  
Unit Cube ◽  
Combinatorial Optimization Problem ◽  
Total Weight ◽  
General Distributions ◽  
Special Case ◽  
Connectivity Threshold

We propose the following model of a random graph on $n$ vertices. Let $F$ be a distribution in $R_+^{n(n-1)/2}$ with a coordinate for every pair $ij$ with $1 \le i,j \le n$. Then $G_{F,p}$ is the distribution on graphs with $n$ vertices obtained by picking a random point $X$ from $F$ and defining a graph on $n$ vertices whose edges are pairs $ij$ for which $X_{ij} \le p$. The standard Erdős-Rényi model is the special case when $F$ is uniform on the $0$-$1$ unit cube. We examine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the $X_{ij}$ are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight.

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 Local Resilience for Squares of Almost Spanning Cycles in Sparse Random Graphs

10.37236/6281 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Andreas Noever ◽  
Angelika Steger
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Hamiltonian Cycle ◽  
Blow Up ◽  
Minimum Degree ◽  
Single Vertex ◽  
Long Cycle ◽  
Local Resilience ◽  
Delta G ◽  
Spanning Cycles

In 1962, Pósa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\delta(G)\ge 2n/3$. Only more than thirty years later Komlós, Sárkőzy, and Szemerédi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $\epsilon > 0$ and $p=n^{-1/2+\epsilon}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+\epsilon)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for $c<2/3$ a.a.s. $G_{n,p}$ contains a subgraph with minimum degree at least $cnp$ which does not contain the square of a path on $(1/3+c)n$ vertices.

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