B. Bollobas. Extremal graph theory with emphasis on probabilistic methods. Regional conference series in mathematics, No. 62, American Mathematical Society, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....8 10 then N g 5 3 g log log g and otherwise N g 7g. 3 The Combinatorial Technique In this section we sketch a combinatorial technique that allows us to construct triangulations of high representativity. For a full presentation we refer the reader to [17] We use a result of Erdos and Sachs [7, 8] and a method similar to that used by Buser [4] To describe the result of Erdos and Sachs we need to introduce a few more definitions. Let G be a graph with at least one cycle. The girth of G is equal to the length of the shortest cycle in G. The number of edges incident with a vertex is called ....

....the shortest cycle in G. The number of edges incident with a vertex is called the valency of the vertex (a loop is counted twice) A graph all of whose vertices have valency 3 is called a cubic graph. surface triangulations 17 The following theorem is a special case of theorem of Erdos and Sachs [7, 8]. Theorem 3.1 ( 7, 8] For any l 2 and for any 2 l Gamma 1 N 2 l Gamma1 Gamma 1 there exists a cubic graph of 2N vertices with girth at least l. Below we sketch the idea of the construction implied by this theorem. Theorem 3.2 ( 17] For any g 2 and any n 4:5g log g f ( Sigma g ; n) ....

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B.Bollob'as, Extremal Graph Theory with Emphasis on Probabilistic Methods, Regional conference series in mathematics, 62 (1986), AMS.

....k lists to the edges and seek to choose a proper edge coloring. Let 0 l (G) be the minimum size of lists needed to guarantee the existence of such a choice function. Equivalently, 0 l (G) l (L(G) where L(G) is the line graph of G. The List Coloring Conjecture, which Bollob as [3] reports is attributed variously to Gupta, Dinitz, Albertson, and Tucker, is Conjecture 3: For ev 2 ery graph G, 0 l (G) 0 (G) The special case 0 l (K n;n ) 0 (K n;n ) was specifically known as the Dinitz Conjecture (1977 see [6] Janssen [20] proved the slightly weaker ....

B. Bollob'as, Extremal Graph Theory with Emphasis on Probabilistic Methods, Chapter 9 (List Colorings), (American Math Society, CBMS Regional Conference Series, Number 62).

.... is equal to width, which is polynomially computable and has an NC approximation algorithm for constant approximation factors 1 2 (see [4] It can be easily proved that width is bounded for classes of graphs with an excluded minor, i.e. graphs with no minor isomorphic to a given graph H (see [17]) However the class of graphs with bounded width is larger: there are graphs with small width containing arbitrary large minors, as is shown in the following lemma. Lemma 3 For any k 3 and any graph H, there is a graph G such that width(G) k and H is a minor of G. Proof Suppose H has h ....

H. Bollob'as, "Extremal graph theory with emphasis on probabilistic methods ", Regional conference series in mathematics, No. 62, American Mathematical Society, 1986.

....cardinality which is an induced subgraph of both G and H . Max Clique Minor (MCM) Given a graph G, find a clique of maximal size which is a minor (i.e. a contraction of a subgraph) of G. The size of such a clique is called the contraction clique number of G, abbreviated ccl(G) It is known [5] that for all almost all graphs, ccl(G) n(log n) Gamma1=2 . Longest Chordless Path (LCP) Given a graph G, find a set V of nodes, as large as possible, such that Gj V is a simple path. Note that to apply the probabilistic criterion to these (and other) problems, it is not necessary to ....

B. Bollob'as, Extremal Graph Theory with Emphasis on Probabilistic Methods, AMS Regional Conference Series in Mathematics, Nr. 62 (1986).

....the graph is d is asymptotically e Gammac=2 and the probability that the diameter of the graph is d 1 is asymptotically 1 Gamma e Gammac=2 . See also Reference [Sp] Randomized algorithms, algorithms that make some random decisions, have been developed for many graph problems, too [Ya87][Bo86]. Their input, behaves much more randomly, and avoids all pathologically bad cases with probability one. Examples of such algorithms follow: ffl The algorithm of Angluin and Valiant that finds Hamilton cycles and matchings and will fail only with probability O(n Gamma2 ) AV] ffl The ....

Bela Bollob'as. Extremal graph theory with emphasis on probabilistic methods, Providence, R.I. : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1986.

....we show a very simple combinatorial construction that not only improves the lower bound for the representativity to Omega p n=g p log g) but also leads to an O(max(g 2 ; n) time algorithm to construct the corresponding triangulation. The key to our solution is a theorem of Erdos and Sachs [4, 5]. This theorem was used before by Buser in a construction of a hyperbolic structure without short geodesics [3] Our approach was 1 In the paper we use log n to denote log 2 n. motivated by the construction of Buser and the idea that in a triangulation with a large edge width, the vertices ....

....equal to = girth(G 0 ) To minimize the number of vertices used in the construction we should start our construction with G 0 being a (3; 0cage. Unfortunately, there is no efficient algorithm that, for a given value , constructs a (3; 0cage. However, by a theorem of Erdos and Sachs [4, 5], for any integer , there exists a cubic graph of girth and O(2 ) vertices which can be constructed efficiently. In Section 5, based on the proof of the theorem of Erdos and Sachs [4, 5] we give an O(N 2 ) time algorithm to construct a cubic graph with 2N vertices and girth 2(log N ) ....

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B.Bollob'as, Extremal Graph Theory with Emphasis on Probabilistic Methods, Regional conference series in mathematics, 62 (1986), AMS.

....d(G) can be found in O(E) time. Let G be an undirected graph. A graph H is a minor of G if it can be obtained from G by the removal and the contraction of edges. A family C of graphs is said to be minor closed if a minor of a graph of the family is also a member of the family. It is known (see [Bol86] p. 7) that if C is a non trivial minor closed family of graphs, i.e. a minor closed family which is not the family of all graphs, then all graphs in C are of bounded degeneracy. In other words, there exists a constant d = d C such that every G 2 C satisfies d(G) d. As an example, consider the ....

B. Bollob'as. Extremal graph theory with emphasis on probabilistic methods. Regional conference series in mathematics, No. 62, American Mathematical Society, 1986.

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B. Bollobas. Extremal graph theory with emphasis on probabilistic methods. Regional conference series in mathematics, No. 62, American Mathematical Society, 1986.

No context found.

B. Bollob'as. Extremal Graph Theory with Emphasis on Probabilistic Methods, volume 62 of Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1986.

No context found.

B. Bollob'as. Extremal Graph Theory with Emphasis on Probabilistic Methods, volume 62 of Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1986.

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