B. Bollob'as, Extremal Graph Theory, Academic Press, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with e(G) Note that this would mean that the upper bound of Theorem 9 gives the correct order of magnitude. The case s = t of the conjecture (which of course would already imply the general case) as well as a proof of the conjecture for the case t s with s = 2, 3 can be found in Bollobas [2]. Based on a construction in [8] Alon, Ronyai and Szabo [1] proved that the conjecture is also true for all t 2 with t (s 1) For s 4 the best known lower bound on the maximum number of edges of a K s,s free graph G is c G 2 2 (s 1) see e.g. 2, Ch. VI, Thm. 2.10] Using this ....

B. Bollobas, Extremal Graph Theory, Academic Press 1978.

....with e(G) Note that this would mean that the upper bound of Theorem 9 gives the correct order of magnitude. The case s = t of the conjecture (which of course would already imply the general case) as well as a proof of the conjecture for the case t s with s = 2, 3 can be found in Bollobas [2]. Based on a construction in [8] Alon, Ronyai and Szabo [1] proved that the conjecture is also true for all t 2 with t (s 1) For s 4 the best known lower bound on the maximum number of edges of a K s,s free graph G is c G 2 2 (s 1) see e.g. 2, Ch. VI, Thm. 2.10] Using this ....

B. Bollobas, Extremal Graph Theory, Academic Press 1978.

....(H) Gamma 1 o(1) 1) Furthermore, as proved independently by Erdos and Simonovits, every H free graph G = G that has as many edges as (1) is in fact very close (in a certain precise sense) to the densest n vertex ( H) Gamma 1) partite graph. For these and related results, see [4, 20, 21]. Thus the situation for graphs H with (H) 3 is quite well understood. Unfortunately, the same cannot be said about the case in which we forbid a bipartite graph H in our G = G . Indeed, in this case, usually referred to as the degenerate case, relation (1) only tells us that ex(n; H) ....

B. Bollob'as. Extremal Graph Theory. Academic Press, London, 1978.

....p) For (2.6) replace Y # by 1 Y # and apply (2.4) Finally, 2.7) is trivial for t EX = Np, and for t Np it follows from (2.6) and (3.12) using (1 p) 1 4t 15N(1 p) p t N 1. Proof of Theorem 2.5. For Theorem 2. 5, we use instead the Hajnal Szemeredi Theorem [12] [5], which says that can be partitioned into # 1 independent sets, each of size equal to either or #. We use these sets as our j , with all weights w j = 1. If N = k# 1 l, with k and l integers and l # 1 , there are necessarily # 1 l sets of size k and l of size k 1, and ....

B. Bollobas, Extremal Graph Theory. Academic Press, London, 1978.

....v) k # , which contradicts (1) Proof of Proposition 5. Let G = V, E) be a graph of diameter 2 on vertices, with no independent sets and no cliques larger than C # log n, here C is an absolute constant. It is well known and easy to prove that almost all graphs have these properties, see [9, 6]. Let M be the metric defined by G. Define M 1 = M , and M i = M # [M i 1 ] where # = 2. First we prove by induction that for each i M i is # embeddable in an equilateral space then C # log n. For i = 1 consider a subset S M that is # embeddable in an equilateral space. Since # 2, and G ....

B. Bollobas. Extremal graph theory. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1978.

....14, 2002 Abstract A graph property is called elusive (or evasive) if every algorithm for testing this property has to read in the worst case entries of the adjacency matrix of the given graph. Several graph properties have been shown to be elusive, e.g. planarity [3] or k colorability [5]. A famous conjecture of Karp says that every non trivial monotone graph property is elusive. We prove that a non monotone but hereditary graph property is elusive: perfectness. 1 Introduction Given a graph property, consider the following two players game to de ne elusiveness. Player A ....

....zu Berlin, Institut f ur Informatik, Unter den Linden 6, D 10099 Berlin, Germany, hougardy informatik.hu berlin.de Konrad Zuse Zentrum f ur Informationstechnik Berlin (ZIB) Takustr. 7, D 14195 Berlin, Germany, wagler zib.de and being planar for graphs on 5 nodes (Best et al. 3] see [5, 11] for more examples. Aanderaa and Rosenberg conjectured [3] that there exists some 0 such that the complexity of every non trivial monotone graph property (i.e. a property preserved under deleting edges) is at least n . This conjecture has been proved by Rivest and Vuillemin for = 16 . ....

B. Bollobas, Extremal Graph Theory, Academic Press (1978)

.... are fixed to constants at a doubly exponential rate, as the computation proceeds, and so it can never obtain a lower bound larger than ##an# log n) To get better bounds, Meyer auf der Heide and Wigderson (1987) 181] apply di#erent Ramsey type techniques, they use Erdos and Rado s theorem [93, 92, 135, 42] to obtain a lower bound## # log n) on sorting n integers. The variables are not fixed, instead some information about their ordering is given to the algorithm as the computation proceeds. This forces a processor to merge ordered sets of variables, and so they modified Valiant s comparison ....

....required for a randomized algorithm, called # test, to distinguish between the case of a sequence of n integers satisfying a certain property, and the case that it has to be modified in more than #n places to make it satisfy the property. He uses the following corollary of the Ramsey theorem [42, 135]: If any finite family of functions with k variables from the positive integers to a finite range, then there exists an infinite subset E of the positive integers, such that the restriction of the members of to E are all order based in their variables. He proves that testing order dependent ....

B. Bollobas, Extremal Graph Theory, Academic Press, New York (1978).

....treewidth and a polynomial time algorithm for computing pathwidth. In Section 5, we present a connection between the parameters of treewidth and degeneracy for k chordal graphs. The degeneracy of a graph G = V; E) is defined to be the maximum min degree of any of the subgraphs of G (see also [16, 20, 31, 35, 36, 41]) In [37] it is proved that the degeneracy of a graph is equal to its width, a graph parameter that is also known as linkage (see also [20, 25, 36] A layout L of a graph G = V; E) is a bijective function, mapping its vertices to numbers f1; 2; jV jg. The width of a layout L of G is ....

H. Bollob'as, "Extremal graph theory", Academic Press, 1978.

....D(I) i.e. s c ISl = and (s,t :s t : i,j k: sl tj) For k]n, we say that a string s D, I) is derived from S e X(I) if s is formed by concatenating different elements from S. Next we review some results from extremal graph theory. The interested reader is referred to the book of Bollobs [9], for background, proofs, etc. Define c(m, l) m, l) to be the maximum number of edges in a directed (undirected) graph with m vertices, that does not contain a cycle with length l, and let fi(m, l) N,l,a I N: a(N,t) N,I) N(N 1) Proof. Let G = V, E) be a directed graph with (N, ....

Bollobas, B., Extremal Graph Theory, Academic Press, London, 1978.

....by Mader [39] however the interested reader may also see [12, Chapter 7, Theorem 1.16] Lemma 4.3. Let p N. There exists a number c p such that for any graph G = V, E) if E # V then K p G. Note: It is known that c p 8(p 2) #log(p 2)#. In Theorem 1. 14 of Chapter 7 of [12] is an easy proof of the weaker result that c p p 3 . There is some evidence that c p = p 2 or at least c p = O(p) however, this is still open. See [12, Page 378] Lemma 4.4. If f is a property of graphs that is closed under minors then, for all G = V, E) such that f(G) 1, E = O( V ....

B. Bollobas. Extremal Graph Theory. Academic Press, New York, 1978.

....D T . Even in the most basic case D T = 2, that is that of verifying whether a graph has a Hamiltonian path, just few results are known. For example, it is known that when a graph is sufficiently dense or its degree sequence satisfies certain conditions, then it has a Hamiltonian path (see, e.g. Bol78] Les96] Fau96] In the case of planar graphs, Tutte [Tut56] showed that every 4 connected planar graph has a Hamiltonian cycle and Chiba and Nishizeki [CN89] provided a linear time algorithm for constructing one. On the opposite side, Garey et al. GJT76] proved that it is N P complete to ....

....is N P complete to decide if it has a D T spanning tree. Additionally we show how the result can be extended to biconnected graphs embeddable into the projective plane, the torus, and the Klein bottle. Basic Definitions This chapter is based on books of White [Whi73] Ringel [Rin74] Bollobas [Bol78] Even [Eve79] Tutte [Tut84] Gross and Tucker [GT87] Bredon [Bre93] Schmidt and Strohlein [SS93] and Henle [Hen94] as well as on articles of Guibas and Stolfi [GS85] Archdeacon [Arc96] and Ellingham [Ell96] In Section 2.1 we introduce basic graph theoretic notations used in this thesis ....

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B. Bollobas. Extremal Graph Theory. Academic Press, London, 1978.

....consequence of Facts 29 and 30. We shall call H the universal b covering of G and denote it by UBC(G) Theorem 33 Let d 2 N . The transduction mapping jGj 2 to jUBC(G)j 1 for connected graphs G of degree at most d is MS compatible. Let G be a graph of degree at most d. By Vizing s theorem (see [1]) there exists an edge coloring of G with m = d 1 colors such that no two adjacent distinct edges have the same color. The result is proved in [1] for finite graphs but the extension to infinite graphs is an easy application of Koenig s lemma (see [3] The coloring can be defined by a partition ....

....jGj 2 to jUBC(G)j 1 for connected graphs G of degree at most d is MS compatible. Let G be a graph of degree at most d. By Vizing s theorem (see [1] there exists an edge coloring of G with m = d 1 colors such that no two adjacent distinct edges have the same color. The result is proved in [1] for finite graphs but the extension to infinite graphs is an easy application of Koenig s lemma (see [3] The coloring can be defined by a partition X 1 ; Xm of EG in m sets. We let X 0 = fsg be a singleton with s 2 VG . We now construct from 30 (G; X 0 ; Xm ) deterministic ....

B. Bollobas. Extremal graph theory. Academic Press, 1978.

....if G is a complete r partite graph with sizes of the maximal independent sets as equal as possible. We will denote this graph by T r (n) or just T r . The Tur an graphs come up in different contexts all over extremal graph theory and are optimal in many senses (one can find a nice survey in [B]) In our case they turn out to determine the extreme points in the convex hulls of the sets of f vectors and fi vectors of flag complexes. Let us denote the f vector of T r (n) by F r (n) or just F r ) and the fi vector by B r (n) or just B r ) Then F r (n) i Gamma1 = X ....

B. Bollobas, Extremal graph theory, Academic Press, 1978.

....we study the behavior of f(n, p, q) between the linear and quadratic orders of magnitude. In particular we show that that we can have at most log p values of q which give a linear f(n, p, q) 1 Introduction 1. 1 Notations and definitions For basic graph concepts see the monograph of Bollobas [1]. V (G)andE(G)denotethe vertex set and the edge set of the graph G. K n is the complete graph on n vertices. In this paper log n denotes the base 2 logarithm. pr(n) denotes the parity of the natural number n,soitis1ifn is odd and 0 otherwise. 1.2 Edge colorings with at least q colors in every ....

B. Bollobas, Extremal Graph Theory, Academic Press, London (1978).

....bound is based on well known results for the Zarankiewicz function. Let z(r, s) denote the maximum number of edges that a subgraph of K r,r can have if it does not contain K s,s as a subgraph. We use the bound z(r; s) # s 1 r # 1 s r(r s 1) s 1)r, 1) which is found in [2] and elsewhere. To prove b(m, n) # r it su#ces to show that z(r; m) z(r; n) r 2 . Take # 0 and set r = c(n log n) m where c = m 1) m 1) 1 #) Then z(r; m) r 2 # m 1 r # 1 m # 1 m 1 r # m 1 r = # m 1 c # 1 m log n n O ## log n n # m # . ....

B. Bollobas, Extremal Graph Theory, in Handbook of Combinatorics, volume II, R. L. Graham, M. Grotschel, and L. Lovasz, eds, MIT Press, Cambridge, Mass., 1995.

....this research was performed during a visit at AT T Labs Research 1. Introduction All graphs considered here are finite, undirected, and have neither loops nor parallel edges; let G denote the family of all such possible graphs with labeled sets of vertices. In what follows, we use the notation of [5] except where stated otherwise; in particular, jGj denotes the number of vertices of a graph G 2 G. Let P be a property of graphs. A graph G with n vertices is called far from satisfying P if no graph G with the same vertex set, which differs from G in no more than n 2 places (i.e. can ....

....(as a function of the other parameters) even when this is not mentioned explicitly. The following corollary, some versions of which appear in various papers applying the Regularity Lemma, is useful in what follows. It is proven by combining Lemma 3. 3 with Turan s Theorem and Ramsey s Theorem (see [5] for their formulation) We omit the details. Corollary 3.4 For every l and there exists = 3:4 (l; such that for every graph G with n 1 vertices there exist disjoint vertex sets W 1 ; W l satisfying: jW i j n. All l 2 pairs are regular. Either all pairs are ....

B. Bollobas, Extremal Graph Theory, Academic Press, New York (1978).

....[39] however the interested reader may also see [12, Chapter 7, Theorem 1.16] Lemma 4.3. Let p # N. There exists a number c p such that for any graph G = V, E) if E # c p V then K p # G. Note: It is known that c p # 8(p 2) #log(p 2)#. In Theorem 1. 14 of Chapter 7 of [12] is an easy proof of the weaker result that c p # 2 p 3 . There is some evidence that c p = p 2 or at least c p = O(p) however, this is still open. See [12, Page 378] Lemma 4.4. If f is a property of graphs that is closed under minors then, for all G = V, E) such that f(G) 1, E = ....

B. Bollobas. Extremal Graph Theory. Academic Press, New York, 1978.

....of Education and Culture of Spain, Grant number MEC DGES SB98 0K148809. x Departament de Llenguatges i Sistemes Inform atics, Universitat Polit ecnica de Catalunya, Campus Nord M odul C5, c Jordi Girona Salgado 1 3, 08034 Barcelona, Spain. Email: sedthilk lsi.upc.es 1 min max theorems see [5]) Several examples of such characterizations emerge from the work of Robertson and Seymour on their Graph Minors series. As a sample, we mention the characterization of treewidth via screens [20] of branchwidth via tangles [19] of pathwidth via blockages [3] and of carving width via ....

B. Bollobas, Extremal Graph Theory, Academic Press, London, 1978.

....For the moment, 9) is limited to values of k such that k = o(n 1=3 ) If this restriction can t be seen as a defect, any improvement of such limitation is of great importance. In fact, motivations include not only asymptotic methods for coecients de ned recursion but extremal graph theory [4], random graphs [5, 7] thresholds characteristics for appearance of forbidden cycles in connected graphs, Since in general, it is quite dicult to extend the range of an asymptotic formula, and it is easier to prove a result if one knows what to start with [1] Our aim in this note is ....

Bollobas, B. Extremal Graph Theory, Academic Press (1978).

....the circuit; with each variable it queries, it pays the associated cost. We could ask for an algorithm A that incurs the minimum worst case cost over all settings of the variables; but this is too simplistic: many of the natural functions we wish to study (including all AND OR trees) are evasive [3], so any algorithm can be made to pay for all the variables, and all algorithms perform equally poorly under this measure. The competitive analysis of algorithms [2] fits naturally within our framework; we define the performance of an algorithm A on a given setting oe of the variables to be the ....

B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

....graph K 3;3 : Similarly, constant size sets of forbidden minors can be used to characterize sparsity graphs arising from various discretizations of 2D PDEs. Fix a (minor closed) family F of graphs defined by a constant size set S F of forbidden graph minors. F is closed under edge contractions [B 78] that is, if we apply a sequence of edge contraction to a graph of F ; the result is a graph in F . Then by [B 78, AST 90] there is some fixed graph that is not the minor of any graph in F and hence the graphs in F are sparse, with m = O(n) edges and they have O( p n) separators. 4 1.7 ....

....from various discretizations of 2D PDEs. Fix a (minor closed) family F of graphs defined by a constant size set S F of forbidden graph minors. F is closed under edge contractions [B 78] that is, if we apply a sequence of edge contraction to a graph of F ; the result is a graph in F . Then by [B 78, AST 90] there is some fixed graph that is not the minor of any graph in F and hence the graphs in F are sparse, with m = O(n) edges and they have O( p n) separators. 4 1.7 Direct Methods The arithmetic bounds for the solution of dense linear systems are known to be within a constant ....

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B. Bollobas, Extremal Graph Theory, Academic Press Inc., (1978).

....converse is proved similarly. 2 Let M be a set of graphs. We de ne FORB(M) as the set of undirected graphs H such that G H for no G 2 M . For example the set of undirected planar graphs can be characterized as the class FORB(K 3;3 ; K 5 ) This is a variant of Kuratowski s theorem (see Bollobas [5]) We recall that Kn is the n clique consisting of n vertices and an edge linking any two distinct vertices, and that K n;m is the complete bipartite graph with n m vertices. Formally its vertices are n; n 1) 1; 1; 2; m and there is an edge linking any negative vertex and any ....

: BOLLOBAS B., Extremal Graph Theory, Academic press, New York, 1978.

.... # D,1 h u (3) For a strongly convex subgraph S of an abelian homogeneous graph #, we show that # D,2 # D,1 # 1 8kD 2 where D denotes the diameter of S and k is the degree of # (which is regular) For undefined terminology in graph theory and spectral geometry, the reader is referred to [4, 17] and [12, 30] respectively. The organizaton of this paper is as follows: In 2 we give basic definitions and describe basic properties for the Laplacian of graphs. In 3 we define a weighted graph Laplacian and its associated first eigenvalue and the weighted Cheeger s constant. In 4 we prove ....

B. Bollobas, Extremal Graph Theory, Academic Press, London (1978).

....passing from any source processor to its desired destination processor. It is important to show how the maximum number of processors that can be interconnected in a configuration of a certain diameter can be derived [4] An upper bound on this value may be obtained using extremal graph theory [5]. Assuming that every processor in the configuration has the same number of links, called its valency Delta, the upper bound on the maximum number of processors in the configuration for a given diameter, usually called the Moore limit, can be found by counting the number of processors which are ....

B. Bollobas, Extremal Graph Theory, Academic Press, London, 1978.

....with each variable it queries, it pays the associated cost. We could ask for an algorithm A that incurs the minimum worst case cost over all settings of the variables; but this is too simplistic: many of the natural functions we wish to study (including all Boolean AND OR trees) are evasive [3], so any algorithm can be made to pay for all the variables, and all algorithms perform equally poorly under this measure. The competitive analysis of algorithms [2] fits naturally within our framework; we define the performance of an algorithm A on a given setting # of the variables to be the ....

B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

....applications. Let F be a family of graphs. Denote by T (n, F, r) the maximum number of edges in a graph on n vertices containing no r isomorphic copies of a member of F . T (n, F, 1) T (n, F ) is just the classical Turan number and is among the most studied parameters in extremal graph theory ([2] Chapter 6 pp. 292 367, 10] 11] Chapter 24 pp.1293 1330) Erdos and Stone [9] and, later, Dirac [8] were the first to raise questions concerning the graphs contained as subgraphs in a graph G on n vertices and T (n, F ) t edges, where t is a positive integer. The Erdos Stone theorem states, ....

....one must have not only a copy of K k but also a copy of the complete k partite graph with side length c(k) log n, and hence, in particular, many copies of K k . Dirac s Theorem states that with T (n, K k ) 1 edges there must exist a copy of K k 1 and hence two copies of K k . Rademacher ([2] p. 301) posed the specific question of determining the minimum number of triangles in a graph on n vertices and T (n, K 3 ) t edges, a problem that was much extended and nearly completely solved years later by Lovasz and Simonovits [17] Our main goal is to present a method to tackle the ....

B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

....with each variable it queries, it pays the associated cost. We could ask for an algorithm A that incurs the minimum worst case cost over all settings of the variables; but this is too simplistic: many of the natural functions we wish to study (including all Boolean AND OR trees) are evasive [3], so any algorithm can be made to pay for all the variables, and all algorithms perform equally poorly under this measure. A B C D cost: 1 4 3 6 Figure 1: A Boolean function with priced inputs The competitive analysis of algorithms [2] fits naturally within our framework; we define the ....

B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

....vertex disjoint induced copies of the complete h partite graph with a vertices in each color class. Mathematics subject classification numbers: 05C70, 05C55. All graphs considered here are finite, undirected, and have neither loops nor parallel edges. The notation here follows the convention of [4] except where stated otherwise. Many asymptotic embedding results have been proven by the Regularity Lemma of Szemeredi [6] over the years. See [5] for a survey. One of the results in this area, that of Alon and Yuster [2] has particular relevance to the following. Theorem 1 ( 2] For every ....

....[1] a variant of the Regularity Lemma which is suitable for dealing with induced subgraphs in very general circumstances is presented and proven. The proof of Theorem 2, however, is immediate from Theorem 1 by use of the following well known Ramsey s Theorem. Theorem 3 (Ramsey s Theorem, see e.g. [4]) For every two positive integers a and b there exists R = R(a, b) such that every graph with R vertices contains either a set with a vertices and no edges between them, or a clique with b vertices. ProofofTheorem2:We choose N # = N (h, c, 1 2 #)whereNis the function defined in Theorem 1, ....

B. Bollobas, Extremal Graph Theory, Academic Press, New York (1978).

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B. Bollob'as, Extremal Graph Theory, Academic Press, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

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B. Bollobas. Extremal graph theory. Academic Press, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, NY, 1978.

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B. Bollobas. Extremal graph theory. Academic Press, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, London, 1978.

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B. Bollobas, Extremal graph theory, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, London, 1978.

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B. Bollobas. Extremal Graph Theory. Academic Press, London, New York, 1978.

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Bela Bollobas. Extremal Graph Theory. Academic Press, 1978.

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B. Bollob'as. Extremal graph theory. Academic Press, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

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B. Bollob'as. Extremal Graph Theory. Academic Press, 1978.

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Bollobas, B., Extremal Graph Theory, L.M.S. Monographs, No. 11, Academic Press, London, 1978.

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B. Bollob'as. Extremal Graph Theory. Academic Press, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

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Bollobas, Extremal Graph Theory, Academic Press, New York (1978).

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B. Bollobas, Extremal Graph Theory, Academic Press 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

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B. Bollobas, Extremal Graph Theory, Academic Press, London, 1978.

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Bollobas. Extremal Graph Theory. Acad. Press, 1978.

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