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We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when . We draw n independent points Xi from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i, j) is added with probability , where is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 408‐420, 2014 相似文献
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Let \begin{align*}n\in\mathbb{N}\end{align*}, 0 <α,β,γ< 1. Define the random Kronecker graph K(n,α,γ,β) to be the graph with vertex set \begin{align*}\mathbb{Z}_2^n\end{align*}, where the probability that u is adjacent to v is given by pu,v =α u ? v γ( 1‐u )?( 1‐v )βn‐ u ? v ‐( 1‐u )?( 1‐v ). This model has been shown to obey several useful properties of real‐world networks. We establish the asymptotic size of the giant component in the random Kronecker graph.© 2011 Wiley Periodicals, Inc. Random Struct. Alg.,2011 相似文献
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We show that a typical d‐regular graph G of order n does not contain an induced forest with around vertices, when n ? d ? 1, this bound being best possible because of a result of Frieze and ?uczak [6]. We then deduce an affirmative answer to an open question of Edwards and Farr (see [4]) about fragmentability, which concerns large subgraphs with components of bounded size. An alternative, direct answer to the question is also given. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 149–156, 2008 相似文献
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Martin Hildebrand 《Random Structures and Algorithms》1996,8(4):301-318
This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc. 相似文献
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Tutte conjectured in 1972 that every 4-edge–connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge–connected graph has an edge orientation in which every in-degree is either 1 or 4. We show that the assertion of the conjecture holds asymptotically almost surely for random 5-regular graphs. It follows that the conjecture holds for almost all 4-edge–connected 5-regular graphs. 相似文献
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We study the threshold for the existence of a spanning maximal planar subgraph in the random graph Gn, p . We show that it is very near p = 1/n? We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n½. 相似文献
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We prove that a.a.s. as soon as a Kronecker graph becomes connected it has a finite diameter. 相似文献
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In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and several similarities and differences are pointed out. For many properties our results for the general case are as strong as the known results for random Cayley graphs and sometimes improve the previously best results for the Cayley case. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 相似文献
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We consider the problem of generating a coloring of the random graph ??n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ colors available, where Δ is the maximum degree of the graph, and we wish to determine the least β = β(p) such that the distribution is close to uniform in O(n log n) steps of the chain. This problem has been previously studied for ??n,p in cases where np is relatively small. Here we consider the “dense” cases, where np ε [ω ln n, n] and ω = ω(n) → ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β(p) than can be achieved using more general techniques. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 相似文献
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Joel Spencer 《Random Structures and Algorithms》1990,1(2):205-214
For each irrational a, 0<a<1, a particular countable graph G is defined which mirrors the asymptotic behavior of the random graph G(n, p) with edge probability p = n?a. 相似文献
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Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A|≥k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b:V(G)?A with ∑v∈V(G)b(v)=0, there always exists a map f:E(G)?A−{0}, such that at each v∈V(G), in A, then G is A-connected. Let Z3 denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z3-connected. In this paper, we proved the following.
(i)
Every 5-edge-connected graph is Z3-connected if and only if every 5-edge-connected line graph is Z3-connected. (ii)
Every 6-edge-connected triangular line graph is Z3-connected. (iii)
Every 7-edge-connected triangular claw-free graph is Z3-connected.
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Matthias Kriesell 《Journal of Graph Theory》2001,36(1):35-51
A noncomplete graph G is called an (n, k)‐graph if it is n‐connected and G − X is not (n − |X| + 1)‐connected for any X ⊆ V(G) with |X| ≤ k. Mader conjectured that for k ≥ 3 the graph K2k + 2 − (1‐factor) is the unique (2k, k)‐graph. We settle this conjecture for strongly regular graphs, for edge transitive graphs, and for vertex transitive graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 35–51, 2001 相似文献
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Tibor Jordn 《Journal of Graph Theory》1999,31(3):179-193
Let A(n, k, t) denote the smallest integer e for which every k‐connected graph on n vertices can be made (k + t)‐connected by adding e new edges. We determine A(n, k, t) for all values of n, k, and t in the case of (directed and undirected) edge‐connectivity and also for directed vertex‐connectivity. For undirected vertex‐connectivity we determine A(n, k, 1) for all values of n and k. We also describe the graphs that attain the extremal values. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 179–193, 1999 相似文献
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Given a group G, the model denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any and any family of groups Gk of order nk for which , a graph with high probability has diameter at most 2 if and with high probability has diameter greater than 2 if . We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erd?s‐Renyi random graphs. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 218–235, 2014 相似文献
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Threshold probabilities for the existence in a random graph on n vertices of a graph isomorphic to a given graph of order Cn and average degree at least three are investigated. In particular it is proved that the random graph G(n, p) on n vertices with edge probability contains a square grid on En/2 vertices. © 1994 John Wiley & Sons, Inc. 相似文献