Luczak, T., Sparse random graphs with a given degree sequence, in A. M. Frieze and T. Luczak (eds.), Proceedings of the Symposium on Random Graphs, Poznan 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a power law as well. 1.3 Previous Work Strictly speaking our model is a special case of random graphs with a given degree sequence for which there is a large literature. For example, Wormald [17] studied the connectivity of graphs whose degrees are in an interval [r, R] where r 3. # Luczak [13] considered the asymptotic behavior of the largest component of a random graph with given degree sequence as a function of the number of vertices of degree 2. His result was further improved by Molloy and Reed [14; 15] They consider a random graph on n vertices with the following degree ....

Tomasz # Luczak, Sparse random graphs with a given degree sequence, Random Graphs, vol 2 (Poznan, 1989.

....a function of the power. Chung and Lu [12, 13] further extend the analysis to random graphs with arbitrary degree distribution. Newman et al. 38] take a similar approach but use di#erent methods of analysis. Other remarkable works in this direction include Molloy and Reed [34, 35] and # Luczak [29]. Certain questions are likely to prove more amenable to analysis using the later approach than the former and vice versa. Thus, the two approaches are complementary. The second approach to modeling power law graphs attempts to model the evolution of such graphs and the manner in which the power ....

Tomasz # Luczak, Sparse random graphs with a given degree sequence, Random Graphs,vol2(Poznan,

....a component on at least ffln vertices with at least ffin cycles, and no other component has more than O(log n) vertices or more than one cycle. This component is referred to as the giant component of G n;M . For more specifics on these two parameters at and around M = 1 2 n see [3] 11] or [14]. In this paper, we are interested in random graphs with a fixed degree sequence where each graph with that degree sequence is chosen with equal probability. Of course, we have to say what we mean by a degree sequence. If the number of vertices in our graph, n is fixed, then a degree sequence is ....

....c , the expected number of minimally 4 chromatic subgraphs of G n;M=cn is exponentially small. This suggests that determining the minimum value of c for which a random graph with cn edges is a.s. 4 chromatic may require more than a study of the subgraphs with minimum degree 3. Recently Luczak [14] showed (among other things) that if G is a random 3 graph on a fixed degree sequence 1 , with no vertices of degree less than 2, and at least Theta(n) vertices of degree greater than 2, then G a.s. has a unique giant component. Our main theorem also generalizes this result. We set Q(D) P ....

T. / Luczak. Sparse Random Graphs With a Given Degree Sequence. Random Graphs Vol. 2 (1992), 165 - 182.

.... Bollob as [5] Fenner and Frieze [8] For r = o(n 1=2 ) such results could have been proved with the help of the models of [16] and [17] In fact this was done, for Hamiltonicity, up to r = o(n 1=5 ) in an unpublished work by Frieze [9] and for r connectivity, up to r n :002 by Luczak [15]. As [13] proves the case where r n 1=2 log n, this implies G r is r connected and Hamiltonian whp 1 for all 3 r n 4. 2 Generating graphs with a xed degree sequence. Let d = d 1 ; d 2 ; d n ) and let 2D = d 1 d 2 d n ) Let G d be the set of simple graphs G with ....

T. Luczak, Sparse random graphs with a given degree sequence, in Random Graphs Vol. 2, (eds. A.M.Frieze and T. Luczak), Wiley, New York (1992) 165-182.

.... (b) is from Robinson and Wormald [23, 24] Bollob as [6] Fenner and Frieze [8] For r = o(n 1=2 ) such results could probably be proved with the help of the models of [19] and [20] In fact this was done, up to r = o(n 1=5 ) in unpublished work by Frieze [9] and for r n :002 by Luczak [17]. We also consider the independence number (G r ) and the chromatic number (G r ) for r 1. Frieze and Luczak [12] showed that for any xed ; 0 there exists r such that if r r n 1=3 then whp (G r ) 2n r (log r log log r 1 log 2) n r : 1) ....

T. Luczak, Sparse random graphs with a given degree sequence, in Random Graphs Vol. 2, (eds. A.M.Frieze and T. Luczak), Wiley, New York (1992) 165-182.

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Luczak, T., Sparse random graphs with a given degree sequence, in A. M. Frieze and T. Luczak (eds.), Proceedings of the Symposium on Random Graphs, Poznan 1989.

No context found.

Tomasz Luczak, Sparse random graphs with a given degree sequence, Random Graphs, vol 2 (Poznan, 1989.

No context found.

T. Luczak, Sparse random graphs with a given degree sequence, in Random Graphs Vol. 2, (eds. A.M.Frieze and T. Luczak), Wiley, New York (1992) 165-182.

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