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Connectivity properties of random subgraphs of the cube

B. Bollobás, Y. Kohayakawa, T. Luczak
1995 Random structures & algorithms (Print)  
Thus Q t is a spanning subgraph of Q n of size t, and Q t is obtained from Q t−1 by the random addition of an edge of Q n not in Q t−1 .  ...  Clearly Q n is an n-connected graph of diameter and radius n. Write M = n2 n−1 = e(Q n ) for the size of Q n . Let Q = (Q t ) M 0 be a random Q n -process.  ...  Part of this work was done while the first and the third authors were visiting the Instituto de Matemática e Estatística, Universidade de São Paulo.  ... 
doi:10.1002/rsa.3240060210 fatcat:zzhjjyi7frb4xmufshwrgurff4

Page 4037 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
This class coincides with that of partial cubes, and has the property that the edges belonging to each 6@-class F form a matching, and G~ F consists of exactly two connected components [P. M.  ...  Important results are as follows: (i) Every graph is the crossing graph of some median graph; (ii) a partial cube G is bi-connected if and only if G# is connected; (iii) a partial cube has a triangle-  ... 

Page 6639 of Mathematical Reviews Vol. , Issue 92m [page]

1992 Mathematical Reviews  
Zbigniew Palka (PL-POZN) 92m:05167 05C80 Bollobas, B. (4-CAMB); Kohayakawa, Y. (4-CAMB); Luczak, T. (4-CAMB) The evolution of random subgraphs of the cube.  ...  Random Structures Algorithms 3 (1992), no. 1, 55-90. This paper presents the results of detailed studies of the critical phase of the evolution of a random cube.  ... 

On the Largest Eigenvalue of a Random Subgraph of the Hypercube

Alexander Soshnikov, Benny Sudakov
2003 Communications in Mathematical Physics  
Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p.  ...  We prove that the largest eigenvalue of the adjacency matrix of G is almost surely λ_1(G)= (1+o(1)) max(Δ^1/2(G),np), where Δ(G) is the maximum degree of G and o(1) term tends to zero as max (Δ^1/2(G),  ...  The first author would like to thank Sergey Gavrilets and Janko Gravner for bringing this problem to his attention and for useful discussions.  ... 
doi:10.1007/s00220-003-0872-y fatcat:bjc27sf3nvbwxbkkagzischqhu

Page 2318 of Mathematical Reviews Vol. , Issue 86f [page]

1986 Mathematical Reviews  
G is a fixed connected regular graph of order n with no triangles. A random subgraph RG(p) of G is obtained by selecting its lines independently and with a common probability p.  ...  The author presents recent developments in the study of such random graphs. The main subjects covered are threshold functions of and the asymptotic distribution of various balanced subgraphs.  ... 

Page 3 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
theorem) and basic properties such as first-order properties and properties of large subsets of vertices.  ...  In the case of circular arc graphs, we show that every integral power of a circular arc graph is a circular arc graph.” 93i:05111 05C75 05E30 Weichsel, Paul M. (1-IL) Distance regular subgraphs of a cube  ... 

Connecting boundary and interior - "Gauss's law" for graphs [article]

Ching King Chan, Kwok Yip Szeto
2011 arXiv   pre-print
The Gauss's law, in an abstract sense, is a theorem that relates quantities on the boundary (flux) to the interior (charge) of a surface.  ...  In this article, we try to construct a definition of flux for other graphs, such that a similar boundary-interior relation can be satisfied.  ...  Szeto acknowledges the support of CERG602507 grant.  ... 
arXiv:1101.5383v1 fatcat:ortwvatgyfdxxowfgo3pxizo6e

Random Induced Subgraphs of Generalizedn-Cubes

Christian M. Reidys
1997 Advances in Applied Mathematics  
ACKNOWLEDGMENTS I gratefully acknowledge the proofreading and help of W. Y. C. Chen and Q. H. Hou. Further I want to thank C. L. Barrett for many discussions.  ...  Finally, the threshold value for the connectivity property P, Q w x of random induced subgraphs 13, 11 is determined.  ...  Ž . vertices of Q Q are n-tuples x , . . . , x , where x g A A, i s 1, . . . , n, and two The main motivation for introducing random induced subgraphs of generalized n-cubes, Q Q n , is the analysis of  ... 
doi:10.1006/aama.1997.0553 fatcat:bn2dwvm7wbewzmgsl6rm7kzgnu

Page 5481 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
Summary: “The subject of this paper is to analyse connectivity and paths in random induced subgraphs of generalized n-cubes @" over an alphabet # with |./| = a.  ...  {For the entire collection see MR 98e:00009. } Martin Baéa (SK-KTU; KoSice) 98i:05146 05C80 Reidys, Christian M. (1-LANL; Los Alamos, NM) Random induced subgraphs of generalized n-cubes.  ... 

Page 6421 of Mathematical Reviews Vol. , Issue 91M [page]

1991 Mathematical Reviews  
Summary: “Dyer, Frieze and Foulds introduced a general model of random subgraphs G,, of the n-cube graph Q,.  ...  (DDR-SEE) On the radius of random subgraphs of the -cube. Topics in combinatorics and graph theory (Oberwolfach, 1990), 505-511, Physica, Heidelberg, 1990.  ... 

Edge Expansion of Cubical Complexes [article]

Thomas Voigt
2005 arXiv   pre-print
In this paper we show that graphs of "neighbourly" cubical complexes -- cubical complexes in which every pair of vertices spans a (unique) cube -- have good expansion properties, using a technique based  ...  By showing that graphs of stable set polytopes are graphs of neighbourly cubical complexes we give a new proof that graphs of stable set polytopes have edge expansion 1.  ...  Acknowledgements I thank Volker Kaibel for many helpful suggestions which led to a significantly simpler proof of Lemma 2.1.  ... 
arXiv:math/0506478v1 fatcat:dzled3pwrvc3rdfh4g4tqkjwae

Fast Approximation Schemes for Euclidean Multi-connectivity Problems [chapter]

Artur Czumaj, Andrzej Lingas
2000 Lecture Notes in Computer Science  
The efficiency of our scheme relies on transformations of nearly optimal low cost special spanners into sub-multigraphs having good decomposition and approximation properties and a simple subgraph connectivity  ...  Furthermore, by showing a low-cost transformation of a k-edge connected graph maintaining the k-edge connectivity and developing novel decomposition properties, we derive a PTAS for Euclidean minimum-cost  ...  Acknowledgments We thank unknown referees for their valuable comments and suggestions (because of space considerations not all of the suggestions could be implemented).  ... 
doi:10.1007/3-540-45022-x_72 fatcat:as6r3t3hfzdajin4xybibstxtq

Virtually Fibering Right-Angled Coxeter Groups [article]

Kasia Jankiewicz, Sergey Norin, Daniel T. Wise
2017 arXiv   pre-print
The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails.  ...  Among the successful examples are the right-angled reflection groups in H^4 with fundamental domain the 120-cell or the 24-cell.  ...  Its vertex-face incidence graph VF(Θ) has the property that any proper connected subgraph with κ ≥ 0 is a square or a chain of length ≤ 2.  ... 
arXiv:1711.11505v1 fatcat:wrf6agvf2bgevf7un2ozlskrvq

Page 4156 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews  
(D-SEFA) On the diameter of random subgraphs of the n-cube. Contemporary methods in graph theory, 507-524, Bibliographisches Inst., Mannheim, 1990.  ...  Let G, = (Vn, En) be a subgraph of the n-cube Q” where V,, is randomly sampled from the vertex set of Q” so that P(x € V,) = Py independently of all other vertices and E,, is randomly sampled from the  ... 

Page 4959 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
We derive bounds on the relationship between size and depth for the components of a nearest-neighbor graph and prove some proba- bilistic properties of the k-nearest-neighbors graph for a random set of  ...  The authors say that they have been able to use the same tech- nique also to obtain NP-completeness results for restricted families of graphs, such as induced subgraphs of the cube or trees.  ... 
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