scholarly journals Quasirandomness in Hypergraphs

10.37236/7537 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Elad Aigner-Horev ◽  
David Conlon ◽  
Hiệp Hàn ◽  
Yury Person ◽  
Mathias Schacht
Keyword(s):  
Random Graph ◽  
Edge Density ◽  
Analytic Framework ◽  
Graph Properties ◽  
Vertex Graph ◽  
Discrete Structures

An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others.In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.

scholarly journals Supersaturation of even linear cycles in linear hypergraphs

2020 ◽  
Vol 29 (5) ◽  
pp. 698-721
Author(s):  
Tao Jiang ◽  
Liana Yepremyan
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Independent Interest ◽  
Multiplicative Constant ◽  
Vertex Graph ◽  
Linear Hypergraphs

AbstractA classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

scholarly journals On the Lower Tail Variational Problem for Random Graphs

2016 ◽  
Vol 26 (2) ◽  
pp. 301-320 ◽  
Author(s):  
YUFEI ZHAO
Keyword(s):  
Variational Problem ◽  
Random Graph ◽  
Random Graphs ◽  
Large Deviation ◽  
Tail Probability ◽  
Edge Density ◽  
Lower Tail ◽  
Lower Tail Probability ◽  
Large Deviation Problem ◽  
Deviation Problem

We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed 0 < δ < 1.Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n−αH (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH > 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.

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 Conflict-Free Colouring of Graphs

2013 ◽  
Vol 23 (3) ◽  
pp. 434-448 ◽  
Author(s):  
ROMAN GLEBOV ◽  
TIBOR SZABÓ ◽  
GÁBOR TARDOS
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Domination Number ◽  
Vertex Graph ◽  
Points Of View

We study the conflict-free chromatic number χCFof graphs from extremal and probabilistic points of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number ann-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős–Rényi random graphG(n,p) and give the asymptotics forp= ω(1/n). We also show that forp≥ 1/2 the conflict-free chromatic number differs from the domination number by at most 3.

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 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 A completion of the proof of the Edge-statistics Conjecture

2004 ◽  
Author(s):  
Lisa Sauerman ◽  
Jacob Fox
Keyword(s):  
Asymptotic Behavior ◽  
Complete Graph ◽  
Extremal Combinatorics ◽  
Single Edge ◽  
Expected Number ◽  
Induced Subgraphs ◽  
Large Graphs ◽  
The Family ◽  
Vertex Graph ◽  
Discrete Structures

Extremal combinatorics often deals with problems of maximizing a specific quantity related to substructures in large discrete structures. The first question of this kind that comes to one's mind is perhaps determining the maximum possible number of induced subgraphs isomorphic to a fixed graph $H$ in an $n$-vertex graph. The asymptotic behavior of this number is captured by the limit of the ratio of the maximum number of induced subgraphs isomorphic to $H$ and the number of all subgraphs with the same number vertices as $H$; this quantity is known as the _inducibility_ of $H$. More generally, one can define the inducibility of a family of graphs in the analogous way. Among all graphs with $k$ vertices, the only two graphs with inducibility equal to one are the empty graph and the complete graph. However, how large can the inducibility of other graphs with $k$ vertices be? Fix $k$, consider a graph with $n$ vertices join each pair of vertices independently by an edge with probability $\binom{k}{2}^{-1}$. The expected number of $k$-vertex induced subgraphs with exactly one edge is $e^{-1}+o(1)$. So, the inducibility of large graphs with a single edge is at least $e^{-1}+o(1)$. This article establishes that this bound is the best possible in the following stronger form, which proves a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn: the inducibility of the family of $k$-vertex graphs with exactly $l$ edges where $0<l<\binom{k}{2}$ is at most $e^{-1}+o(1)$. The example above shows that this is tight for $l=1$ and it can be also shown to be tight for $l=k-1$. The conjecture was known to be true in the regime where $l$ is superlinearly bounded away from $0$ and $\binom{k}{2}$, for which the sum of the inducibilities goes to zero, and also in the regime where $l$ is bounded away from $0$ and $\binom{k}{2}$ by a sufficiently large linear function. The article resolves the hardest cases where $l$ is linearly close to $0$ or close to $\binom{k}{2}$, and provides generalizations to hypergraphs.

scholarly journals Local Resilience and Hamiltonicity Maker–Breaker Games in Random Regular Graphs

2010 ◽  
Vol 20 (2) ◽  
pp. 173-211 ◽  
Author(s):  
SONNY BEN-SHIMON ◽  
MICHAEL KRIVELEVICH ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Regular Graph ◽  
High Probability ◽  
Perfect Matching ◽  
Winning Strategy ◽  
Graph Model ◽  
Vertex Connectivity ◽  
Random Graph Model ◽  
Graph Properties ◽  
Local Resilience

For an increasing monotone graph propertythelocal resilienceof a graphGwith respect tois the minimalrfor which there exists a subgraphH⊆Gwith all degrees at mostr, such that the removal of the edges ofHfromGcreates a graph that does not possess. This notion, which was implicitly studied for somead hocproperties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model(n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-randomregulargraphs of constant degree. We investigate the local resilience of the typical randomd-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values ofd, with high probability, the local resilience of the randomd-regular graph,n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model(n, p), for every positive ϵ > 0 and large enough values ofK, ifp>$\frac{K\ln n}{n}$then, with high probability, the local resilience of(n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that ifdis large enough then, with high probability, a typical randomd-regular graphGis such that, in the unbiased Maker–Breaker game played on the edges ofG, Maker has a winning strategy to create a Hamilton cycle.

scholarly journals On the edit distance function of the random graph

2021 ◽  
pp. 1-23
Author(s):  
Ryan R. Martin ◽  
Alex W. N. Riasanovsky
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Distance Function ◽  
Edit Distance ◽  
Golden Ratio ◽  
Edge Density ◽  
Hereditary Property ◽  
Maximum Proportion ◽  
Speed Function ◽  
Vertex Disjoint

Abstract Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$ , the edit distance function $\textrm{ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $\mathcal{H}$ . The edit distance function is directly related to other well-studied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$ -chromatic number of a random graph. Let $\mathcal{H}$ be the property of forbidding an Erdős–Rényi random graph $F\sim \mathbb{G}(n_0,p_0)$ , and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$ , then a.a.s. as $n_0\to\infty$ , \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$ . A primary tool in the proof is the categorization of p-core coloured regularity graphs in the range $p\in[1-1/\varphi,1/\varphi]$ . Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques.

scholarly journals On the Number of Non-Zero Elements of Joint Degree Vectors

10.37236/6385 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Johannes Rauh ◽  
Kayvan Sadeghi ◽  
Taylor Short ◽  
László Székely
Keyword(s):  
Random Graph ◽  
Upper Bound ◽  
Graph Model ◽  
Upper Bounds ◽  
Exponential Random Graph Model ◽  
Lower And Upper Bounds ◽  
Sufficient Statistics ◽  
Random Graph Model ◽  
Exponential Random Graph ◽  
Vertex Graph

Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector as a function of $n$. This provides an upper bound on the number of estimable parameters in the exponential random graph model with bidegree-distribution as its sufficient statistics.

Sign in / Sign up

Export Citation Format

Share Document