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Regular graphs with linearly many triangles [article]

Pim van der Hoorn, Gabor Lippner, Elchanan Mossel
2021 arXiv   pre-print
We compute the leading asymptotics of the probability that a large random d-regular graph has at least c · T_max triangles, and provide a strong structural description of such graphs.  ...  A d-regular graph on n nodes has at most T_max = n/3d2 triangles.  ...  : it is a collection of disjoint d + 1-cliques and an almost triangle-free graph.  ... 
arXiv:1904.02212v3 fatcat:i6xmqhdj4ba7vf6cq7pmjolzgq

Triangle-Free Subgraphs of Random Graphs

2017 Combinatorics, probability & computing  
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 −1 n)-close to bipartite, and each spanning triangle-free subgraph  ...  This motivates the question of which additional restrictions on the class of triangle-free graphs allow for a bound on the chromatic number.  ... 
doi:10.1017/s0963548317000219 fatcat:cok2js2kyjb3xhp3d5n3pqquye

Page 5815 of Mathematical Reviews Vol. , Issue 96j [page]

1996 Mathematical Reviews  
Graph Theory 21 (1996), no. 2, 137-151. In this paper the authors study the asymptotic structure of graphs in the class of triangle-free graphs on n vertices having m = m(n) edges.  ...  Bagaev (Moscow) 96j:05096 05C80 05C75 Prémel, Hans Jiirgen (D-HUMB-II; Berlin); Steger, Angelika (D-DUIS; Duisburg) On the asymptotic structure of sparse triangle free graphs. (English summary) J.  ... 

Analysis of Iterated Greedy Heuristic for Vertex Clique Covering

David Chalupa, Jiří Pospíchal
2018 Computing and informatics  
We show that for triangle-free graphs, IG solves CCP optimally in expected polynomial time.  ...  Secondly, we show that IG finds the optimum for CCP in a specific case of sparse random graphs in expected polynomial time with high probability.  ...  Acknowledgement This contribution was supported by Grant Agency VEGA SR under the grant 1/0145/18.  ... 
doi:10.4149/cai_2018_2_385 fatcat:5qgbyhsu7rbsdklpzqbezipwty

Asymptotics for sparse exponential random graph models

Mei Yin, Lingjiong Zhu
2017 Brazilian Journal of Probability and Statistics  
We study the asymptotics for sparse exponential random graph models where the parameters may depend on the number of vertices of the graph.  ...  They are in sharp contrast to the corresponding asymptotics in dense exponential random graph models.  ...  Acknowledgements The authors are very grateful to the anonymous referee for the invaluable suggestions that greatly improved the quality of this paper.  ... 
doi:10.1214/16-bjps319 fatcat:ovemusqdoncxdiabh6v4ep3mn4

Regular Graphs with Many Triangles are Structured

Pim Van der Hoorn, Gabor Lippner, Elchanan Mossel
2022 Electronic Journal of Combinatorics  
We compute the leading asymptotics of the logarithm of the number of $d$-regular graphs having at least a fixed positive fraction $c$ of the maximum possible number of triangles, and provide a strong structural  ...  When $d$ is constant, we show that such graphs typically consist of many disjoint $(d+1)$-cliques and an almost triangle-free part.  ...  Acknowledgements The authors thank Dmitri Krioukov for useful discussions on the related topic of sparse maximum entropy graphs with given number of triangles, which lead us to the upper tail problem.  ... 
doi:10.37236/10369 fatcat:v3onvwhvxvae3olhdkgamlp7ue

Fast Graphlet Transform of Sparse Graphs [article]

Dimitris Floros and Nikos Pitsianis and Xiaobai Sun
2020 arXiv   pre-print
We introduce the computational problem of graphlet transform of a sparse large graph. Graphlets are fundamental topology elements of all graphs/networks.  ...  They can be used as coding elements to encode graph-topological information at multiple granularity levels for classifying vertices on the same graph/network as well as for making differentiation or connection  ...  We thank the reviewer who suggested the inclusion of experimental timing results and code release in the revised manuscript. We also thank Tiancheng Liu for helpful comments.  ... 
arXiv:2007.11111v3 fatcat:s26ajxepc5h2nmdc7btqi3ptai

The method of hypergraph containers [article]

József Balogh, Robert Morris, Wojciech Samotij
2018 arXiv   pre-print
We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics  ...  In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures.  ...  graphs and on extremal and Ramsey properties of random discrete structures inspired and influenced our investigations that led to the hypergraph container theorems.  ... 
arXiv:1801.04584v1 fatcat:2i524zhma5gbloc7ovoiq2znmi

Special Issue of Journal of Statistical Physics Devoted to Complex Networks

Diego Garlaschelli, Remco van der Hofstad, Frank den Hollander, Michel Mandjes
2018 Journal of statistical physics  
It is a crossroad of concepts, ideas and techniques that bring together scientists from different disciplines, all facing the major challenges that arise from dealing with the measurement, simulation,  ...  modelling and analysis of (typically very large) real-world net-B Frank den Hollander  ...  It is shown that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution.  ... 
doi:10.1007/s10955-018-2166-y fatcat:nsqkugzycbbr5prm3fknmvd5b4

Structural Transitions in Densifying Networks

R. Lambiotte, P. L. Krapivsky, U. Bhat, S. Redner
2016 Physical Review Letters  
Further, there is an infinite sequence of structural anomalies at p=2/3, 3/4, 4/5, etc., where the dependences on N of the number of triangles (3-cliques), 4-cliques, undergo phase transitions.  ...  The resulting network is sparse for p<1/2 and dense (average degree increasing with number of nodes N) for p≥1/2.  ...  A more relevant criterion is not defect-free completeness, but whether the number of links eventually scales as N 2 /2, as in the complete graph.  ... 
doi:10.1103/physrevlett.117.218301 pmid:27911534 fatcat:xb7xzrjyyngbzd5jiyldmyl4ai

The educational insights and opportunities afforded by the nuances of Prim's and Kruskal's MST algorithms

Ali Erkan
2018 Proceedings of the 23rd Annual ACM Conference on Innovation and Technology in Computer Science Education - ITiCSE 2018  
Our study also includes the performance consequences of the qualities of the graphs on which MSTs are computed: structure (random vs scale-free), density (sparse vs dense), and edge cost distribution.  ...  For a graph of n nodes and e edges, both run in O (e lg n) time but these results are based on picking the right data structures.  ...  node's edge count; this structure is known as a scale-free graph in the literature.  ... 
doi:10.1145/3197091.3197129 dblp:conf/iticse/Erkan18 fatcat:de27vwplyzf3dowscwsieyqxjm

Palette Sparsification Beyond (Δ+1) Vertex Coloring

Noga Alon, Sepehr Assadi, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
between (1+ε) Δ and (Δ+1) coloring in the context of palette sparsification. - A natural family of graphs with chromatic number much smaller than (Δ+1) are triangle-free graphs which are O(Δ/ln Δ) colorable  ...  We prove a palette sparsification theorem tailored to these graphs: Sampling O(Δ^γ + √{log n}) colors per vertex is sufficient and necessary to obtain a proper O_γ(Δ/ln Δ) coloring of triangle-free graphs  ...  While the exact implementation of this technique varies significantly from one application to another, the basic idea is as follows: Partition the vertices of the graph G randomly into multiple parts V  ... 
doi:10.4230/lipics.approx/random.2020.6 dblp:conf/approx/AlonA20 fatcat:77ofth4lufccjeu2uawgh4w56a

Lack of Hyperbolicity in Asymptotic Erdös--Renyi Sparse Random Graphs [article]

Onuttom Narayan, Iraj Saniee, Gabriel H. Tucci
2012 arXiv   pre-print
In this work we prove that the giant component of the Erdös--Renyi random graph G(n,c/n) for c a constant greater than 1 (sparse regime), is not Gromov δ-hyperbolic for any positive δ with probability  ...  tending to one as n→∞.  ...  On the other hand, the Cayley graph associated with the product of two free groups, G = F 2 × F 2 , has a positive Cheeger constant and non-zero spectral gap.  ... 
arXiv:1009.5700v2 fatcat:nqvj2dw5pbhrlfk6dpboyosupa

Turan numbers for bipartite graphs plus an odd cycle [article]

Peter Allen, Peter Keevash, Benny Sudakov, Jacques Verstraete
2012 arXiv   pre-print
Our general approach to the Erdős-Simonovits conjecture is effective based on some reasonable assumptions on the maximum number of edges in an m by n bipartite F-free graph.  ...  K_2,t-free bipartite graph on n vertices.  ...  Similarly, our example of a dense K 2,3 -free triangle-free graph in Theorem 1.6 is actually a triple of sparse-regular pairs, so the cluster graph is actually a triangle, whereas the original graph has  ... 
arXiv:1210.3805v1 fatcat:g3g4dwcpjfhz7ltlvl66vt57ci

Page 8631 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
In particular, it is shown how the Kleitman-Rothschild method can be applied to prove that almost all triangle-free graphs are 2-colourable or to show that almost all posets are 3-layer posets.  ...  Summary: “We propose a random graph model which is a special case of sparse random graphs with given degree sequences which satisfy a power law.  ... 
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