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Random Cyclic Triangle-Free Graphs of Prime Order

Yu Jiang, Meilian Liang, Yanmei Teng, Xiaodong Xu, Efthymios G. Tsionas
2021 Journal of Mathematics  
Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process.  ...  It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the  ...  A Method on Generating Random Cyclic Triangle-Free Graphs of Prime Order. Suppose that p is an odd prime and p > 120.  ... 
doi:10.1155/2021/5595919 fatcat:k6kao5ulkrhihbedcmhslmjr6u

Triangle-Free Subgraphs of Random Graphs

Peter Allen, Julia Böttcher, Barnaby Roberts, Yoshiharu Kohayakawa
2015 Electronic Notes in Discrete Mathematics  
The Andrásfai-Erdős-Sós Theorem [2] states that all triangle-free graphs on n vertices with minimum degree strictly greater than 2n/5 are bipartite.  ...  We prove best possible random graph analogues of these theorems.  ...  This motivates the question of which additional restrictions on the class of triangle-free graphs allow for a bound on the chromatic number.  ... 
doi:10.1016/j.endm.2015.06.055 fatcat:wvqdi4j3knd7bfyrn7tkmgafoq

Triangle-Free Subgraphs of Random Graphs

PETER ALLEN, JULIA BÖTTCHER, YOSHIHARU KOHAYAKAWA, BARNABY ROBERTS
2017 Combinatorics, probability & computing  
Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory.  ...  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

No Dense Subgraphs Appear in the Triangle-free Graph Process [article]

Stefanie Gerke, Tamás Makai
2010 arXiv   pre-print
Consider the triangle-free graph process, which starts from the empty graph on n vertices and a random ordering of the possible n 2 edges; the edges are added in this ordering provided the graph remains  ...  We will show that there exists a constant c such that no copy of any fixed finite triangle-free graph on k vertices with at least ck edges asymptotically almost surely appears in the triangle-free graph  ...  In this paper we show that there exists a constant c such that a.a.s. no copy of any fixed finite triangle-free graph on k vertices with at least ck edges appears in the triangle-free graph process.  ... 
arXiv:1002.2316v1 fatcat:hkihzcup2reoblgtwegdnl7hd4

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.  ...  Therefore, 1-cliques actually perform fair random walks on the triangle-free graph.  ... 
doi:10.4149/cai_2018_2_385 fatcat:5qgbyhsu7rbsdklpzqbezipwty

On the average size of independent sets in triangle-free graphs

Ewan Davies, Matthew Jenssen, Will Perkins, Barnaby Roberts
2017 Proceedings of the American Mathematical Society  
We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree d.  ...  We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on n vertices with maximum degree d, showing that an independent set drawn uniformly at random  ...  Acknowledgements We thank Rob Morris for his helpful comments on this paper, Noga Alon and Benny Sudakov for recalling James Shearer's 1998 SIAM talk, and James Shearer for sharing his transparencies from  ... 
doi:10.1090/proc/13728 fatcat:ij5ofdujgfeuvpg3t6zxgtr3v4

Testing perfection is hard [article]

Noga Alon, Jacob Fox
2011 arXiv   pre-print
One of our main results shows that testing perfectness is hard. The proof shows that testing perfectness is at least as hard as testing triangle-freeness, which is hard.  ...  A graph property P is strongly testable if for every fixed ϵ>0 there is a one-sided ϵ-tester for P whose query complexity is bounded by a function of ϵ.  ...  Theorem 3.1 Let T be a graph on n vertices which is 14ǫ-far from being triangle-free such that a random sample of d vertices of T is triangle-free with probability at least 1/2.  ... 
arXiv:1110.2828v1 fatcat:z75wuesgcra3pjwwboczw2lrte

Uncountable Graphs and Invariant Measures on the Set of Universal Countable Graphs [article]

F.V.Petrov, A.M.Vershik
2009 arXiv   pre-print
Such measures can be regarded as random graphs (respectively, random K_s-free graphs).  ...  For the case of the universal K_s-free graphs there were no previously known examples of the invariant measures on the space of such graphs.  ...  Random graphs and invariant measures on the set of the universal graphs We consider a "random K s -free graph," which is the same as a S N -invariant measure on the set of universal K s -free graphs.  ... 
arXiv:0804.3386v3 fatcat:tfchefjsh5cozd4ifbv5gkhuby

Removal lemmas and approximate homomorphisms [article]

Jacob Fox, Yufei Zhao
2022 arXiv   pre-print
One such variant, which we call the triangle-free lemma, states that for each ϵ>0 there exists M such that every triangle-free graph G has an ϵ-approximate homomorphism to a triangle-free graph F on at  ...  One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in ϵ^-1.  ...  We thank Yuval Wigderson and the anonymous reviewer for careful readings and comments on the manuscript.  ... 
arXiv:2104.11626v2 fatcat:fotjgjbczneyfplbeadibkbble

Easily Testable Graph Properties

NOGA ALON, JACOB FOX
2015 Combinatorics, probability & computing  
{P}$by inspecting the induced subgraph on a random subset of at mostf(ε) vertices.  ...  One of our main results shows that testing perfectness is hard. The proof shows that testing perfectness is at least as hard as testing triangle-freeness, which is hard.  ...  Theorem 3.1 If there is a graph T be a graph on n vertices which is 14 -far from being triangle-free such that a random sample of d vertices of T is triangle-free with probability at least 1/2, then there  ... 
doi:10.1017/s0963548314000765 fatcat:725vmi5afbgnbejcpf2hooyfs4

Large Cuts with Local Algorithms on Triangle-Free Graphs [article]

Juho Hirvonen, Joel Rybicki, Stefan Schmid, Jukka Suomela
2014 arXiv   pre-print
We study the problem of finding large cuts in d-regular triangle-free graphs.  ...  As a corollary, this shows that in any d-regular triangle-free graph there exists a cut of at least this size.  ...  Regular Triangle-Free Graphs In general graphs, we cannot expect to find cuts that are much better than uniform random cuts.  ... 
arXiv:1402.2543v1 fatcat:wvdyr5liwfasbp5p7abmimbjtu

ERGMs are Hard [article]

Michael J. Bannister and William E. Devanny and David Eppstein
2014 arXiv   pre-print
This model represents a probability distribution on graphs by setting the log-likelihood of generating a graph to be a weighted sum of feature counts.  ...  Furthermore, there is no randomized polynomial time algorithm for generating random graphs whose distribution is within total variation distance 1-o(1) of a given ERGM.  ...  Preliminaries Exponential random graphs. An exponential family random graph model, or ERGM for short, is a distribution on random graphs that forms an exponential family.  ... 
arXiv:1412.1787v1 fatcat:67npqnw3g5dy5loyhqn732skzu

A Graph-Theory Framework for Evaluating Landscape Connectivity and Conservation Planning

EMILY S. MINOR, DEAN L. URBAN
2008 Conservation Biology  
Furthermore, the habitat network had commonalities with planar networks, which exhibit slow movement, and scale-free networks, which are resistant to random disturbances.  ...  We introduced graph measures such as compartmentalization and clustering, which can be used to identify locations on the landscape that may be especially resilient to human development or areas that may  ...  Moilanen, and the anonymous reviewers for their comments on this manuscript. Literature Cited  ... 
doi:10.1111/j.1523-1739.2007.00871.x pmid:18241238 fatcat:tp4tafh5nbcofjseyhunougn5q

Large Cuts with Local Algorithms on Triangle-Free Graphs

Juho Hirvonen, Joel Rybicki, Stefan Schmid, Jukka Suomela
2017 Electronic Journal of Combinatorics  
$-regular triangle-free graph.  ...  Let $G$ be a $d$-regular triangle-free graph with $m$ edges.  ...  Fix a triangle-free d-regular graph G = (V, E).  ... 
doi:10.37236/6862 fatcat:3n722m5z4rgavbpnmmog7fkriq

No Dense Subgraphs Appear in the Triangle-free Graph Process

Stefanie Gerke, Tamás Makai
2011 Electronic Journal of Combinatorics  
Consider the triangle-free graph process, which starts from the empty graph on $n$ vertices and in every step an edge is added that is chosen uniformly at random from all non-edges that do not form a triangle  ...  We will show that there exists a constant $c$ such that asymptotically almost surely no copy of any fixed finite triangle-free graph on $k$ vertices with at least $ck$ edges appears in the triangle-free  ...  In this paper we show that there exists a constant c such that, given any fixed finite triangle-free graph on k vertices with at least ck edges, a.a.s. no copy of it appears in the triangle-free graph  ... 
doi:10.37236/655 fatcat:rlm4zjopbnelpmzglfylpwtp6y
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