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Global Connectivity and Expansion: long cycles and.. - Brandt, Broersma.. (2002) (Correct)
....and let B be the set of all the other vertices of G S. Then but S #k 2 2#k 2##2 A , contradicting the f connectedness assumed for G. # For 2 factors we use Tutte s f factor theorem [23, 24] instead of his 1 factor theorem. Readable secondary sources for the f factor theorem include [5] and [18] Proof of Theorem 3.1. By Tutte s f factor theorem, G = V,E) has a 2 factor if and only if for all disjoint subsets A, B V #G (A, B) 2 A v#B dG A (v) 2 B oddG (A, B) 0. Here, odd G (A, B) denotes the number of oddcomponents with respect to (A, B) which are those ....
B. Bollob as, Extremal Graph Theory , Academic Press 1978.
Decomposition if Complete Graphs into k Factors of Diameter 2 - Chen, Füredi, Nedela (Correct)
....diameter 1 is a complete graph. For d i 2, 1 i k, de ne f(d 1 , d 2 , d k ) to be the minimum number n, such that the complete graph K n can be decomposed into k factors, F 1 , F 2 , F k with diam(F i ) d i . This problem can be found in Bollob as classic Extremal Graph Theory [Bol]. J. Bos ak, A. Rosa and S. Zn am [BRZ] proved that if m f(d 1 , d 2 , d k ) then Km can also be decomposed into factors, F 1 , F 2 , F k with diam(F i ) d i . Palumbiny [P] proved that if d 1 , d 2 , d k 3, then f(d 1 , d 2 , d k ) 2k. It is easy to see that for ....
....2k into k factors of diameter 3. Let f(k) denote the minimum n such that K n can be decomposed into k factors of diameter 2. For small k, Bos ak, Rosa and Zn am [BRZ] showed 11 f(3) 13. Stacho and Urland [SU] proved that K 12 cannot be decomposed into three factors of diameter 2. From [BRZ] [Bol], N] it is known that 17 f(4) 24; that 22 f(5) 30 and that 28 f(6) 36. For general k, Bos ak, Erd os and Rosa proved in [BER] that f(k) is nite for all k 2, and they obtained also for the rst bound, f(k) 4:9 k log k, for all suciently large k. Sauer [S] proved that f(k) ....
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B. Bollobas, Extremal Graph Theory, Academic, London, 1978.
Article 03.1.4 - Journal Of Integer (Correct)
....graph on m vertices not containing a complete graph of s vertices, K s , has at most s ) vertices. Proposition 3.1 (Tur an[1, 25] Let 2 m; s be positive integers, then the following are equal. 1. m i s 1 c , see [6, p.294] 7, p.54] 2. 0 i j s 1 m i m j , see [6, 294], 19, p.1234] 3. s 2) m 2(s 1) where k = mod (m; s 1) m (s 1)b s 1 c, see [21, p.18] 14 4. ex(m; K s ) the maximum number of 2 sets (edges) of f1; mg which have no s cliques. s ) sequence snm 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 ....
B. Bollobas, Extremal Graph Theory, Academic Press, 1978. 14
An Information-Theoretic Upper Bound of Planar Graphs.. - Bonichon, Gavoille.. (2003) (3 citations) (Correct)
.... 1) n k 1) 6 n q(n) 1) as q(n) clearly increases with n. So to prove Theorem 5, it remains to prove that, for every n large enough, log 2 q(n) 6 n O(logn) for some constant 5:007: 2) 28 4. 1 Proofs of Theorem 4 and Theorem 5 We have the well known upper bound (see for instance [Bol78, p. 255] for all x y 0 where r = y=x: y x y x y rx rx x rx x rx r r 1 r 1 r x = 2 H(r) x where H is the entropy function of r, de ned for all r 2 [0; 1] by: H(r) r log 2 r (1 r) log 2 (1 r) and with H(0) H(1) 0 : So that, for all ....
Bela Bollobas. Extremal Graph Theory. Academic Press, New York, 1978.
Large Planar Subgraphs in Dense Graphs - Kühn, Osthus, Taraz (Correct)
....1 there exists n 0 = n 0 ( such that every graph G of order n n 0 and minimum degree 1500 n= contains a planar subgraph with at least (2 )n edges. This is essentially best possible in two ways. Firstly, there are graphs with minimum degree n=2 and girth at least 6 ( 6] see also [3]) Hence Euler s formula shows that any planar subgraph of such a graph can have at most (n 2) edges (as all of its facial cycles have length at least 6) Secondly, for n=2 consider the graph consisting of n=2 disjoint copies of the complete bipartite graph K ; It obviously has minimum ....
B. Bollobas, Extremal Graph Theory, Academic Press 1978.
An Information Upper Bound of Planar Graphs Using.. - Bonichon, Gavoille.. (2002) (1 citation) (Correct)
....it turns out p(n) 6 k=1 k q(n k 1) n k 1) 6 n q(n) 1) as q(n) clearly increases with n. So to prove Theorem 5, it remains to prove that, for every n large enough, log 2 q(n) 6 n O(log n) for some constant 5:007: 2) We have the well known upper bound (see for instance [Bol78, p. 255] for all x y 0 where r = y=x: y x y x y rx rx x rx x rx r r 1 r 1 r x H(r) x where H is the entropy function of r, de ned for all r 2 [0; 1] by: H(r) r log 2 r (1 r) log 2 (1 r) and with H(0) H(1) 0 : So that, for all n; ....
Bela Bollobas. Extremal Graph Theory. Academic Press, New York, 1978.
Geometric Graphs With No Self-Intersecting Path of Length.. - Pach, Pinchasi, Tardos.. (2002) (Correct)
.... 2) n 3) provided that k = 1; 2; 3, and the maximum is O( kn) for large values of k (cf. PT97] For a survey of many similar results in Geometric Graph Theory, consult [P99] The above questions can also be regarded as geometric analogues of the fundamental problem of Extremal Graph Theory [B78]: determine the maximum number of edges of all K free graphs on n vertices, i.e. all graphs which do not contain a subgraph isomorphic to a xed graph K. Denote this maximum by ex(n; K) In the present note, we consider the special instance of the above question when H consists of all ....
....length 4. Then G can be drawn in the plane as an x monotone topological graph with the property that any two edges belonging to a path of length 3 cross an even number of times. It is well known that there are C 4 free bipartite graphs of n vertices and at least constant times edges (see e.g. [B78]) In Section 7, we consider geometric and x monotone topological graphs with no self intersecting path of length ve. In this case, Theorem 9 provides a slightly stronger bound on the number of edges than those obtained for graphs with no self intersecting P 3 . We do not believe that Theorem 9 ....
B. Bollobas, Extremal Graph Theory, Academic Press, New York, 1978.
Geometric Graph Theory - Pach (1999) (7 citations) (Correct)
....graph G is a geometric subgraph of another geometric graph H if V (G) V (H) and E(G) E(H) The systematic study of geometric graphs was initiated by S. Avital and H. Hanani [7] P. Erd os, Y. Kupitz [48] and M. Perles. They realized that many classical questions in extremal graph theory [10] have natural analogues for geometric graphs. Some of these questions turned out to be surprisingly dicult and required new techniques combining geometric and combinatorial tools. In this paper we survey some recent results of this type and some tantalizing open problems. Section 2 focuses on ....
B. Bollobas, Extremal Graph Theory , Academic Press, New York (1978).
A Turán-type theorem on chords of a convex polygon - Capoyleas, Pach (2001) (Correct)
....The maximum number of straight line segments connecting n points in convex position in the plane, so that no k 1 of them are pairwise crossing is ( 2 ) if n 2k 1 and 2kn ( 2k 1 2 ) if n 2k 1. 1 Introduction One of the classical results in graph theory is Tur an s theorem (see [T, B]) according to which the maximum number of edges of a graph with n vertices containing no complete subgraph on k 1 vertices is (n ) 2 ) where r is the remainder of n upon division by k. As far as we know, Paul Erd os was the rst to suggest that similar questions can be raised ....
B. Bollobas, Extremal Graph Theory, Academic Press, London-New York, 1978.
Geometric Graphs With No Self-Intersecting Path of Length.. - Pach, Pinchasi, Tardos (2002) (Correct)
.... 2) n 3) provided that k = 1; 2; 3, and the maximum is O( kn) for large values of k (cf. PT97] For a survey of many similar results in Geometric Graph Theory, consult [P99] The above questions can also be regarded as geometric analogues of the fundamental problem of Extremal Graph Theory [B78]: determine the maximum number of edges of all K free graphs on n vertices, i.e. all graphs which do not contain a subgraph isomorphic to a xed graph K. Denote this maximum by ex(n; K) In the present note, we consider the special instance of the above question when H consists of all ....
....4. Then G can be drawn in the plane as an x monotone topological graph with the property that any two edges belonging to a path of length 3 cross an even number of times. It is well known that there are C 4 free bipartite graphs of n vertices and at least constant times n edges (see e.g. [B78]) 2 A Davenport Schinzel bound for double arrays In this section, we discuss the special case of Theorem 1.1 when G is a bipartite geometric (or x monotone topological) graph, whose vertices are divided by the y axis into two classes, A and B, and all edges of G run between these classes. We ....
B. Bollobas, Extremal Graph Theory, Academic Press, New York, 1978.
A Random Walk in Statistical Physics - Svenson (Correct)
....The number of timeslots that are available corresponds to the number of colours allowed to colour the graph. Many interesting results can be shown concerning graph colouring. For instance, it is known that there exists graphs which are not colourable with three colours yet contain no triangles [63]. These graphs are actually quite simply constructed using induction. The first example is a graph with one node and zero edges. This graph is not colourable using 0 colours and contains no triangles. Now assume that there exists graphs not colourable using 0, 1, k colours; we want to ....
B. Bollob as, Extremal Graph Theory. Academic Press, New York, 1978.
Covering and Spanning Tree of Graphs - Bottreau (Correct)
....denote by the reAEexive and transitive closure of Gamma . Denition 4 Let G be a connected graph and A be a spanning tree of G. The spanning tree A is a p spanning tree of G if A G, for p 1. The circumference of a graph G, denoted c(G) is the maximal length of a cycle of G (cf. [Bol78]) Proposition 3 Let G be a connected graph. If the circumference of G is at most p 1, then each spanning tree of G is a p spanning tree. Proof: Consider A a spanning tree of G. We denote by C e the cycle obtained by adding e to A with e 2 E(G) n E(A) As c(G) p 1, for any edge e 2 E(G) n ....
B. Bollob#s. Extremal graph theory. Academic Press, 1978.
Threshold Functions for the Bipartite Turán Property - Godbole, Lamorte, Sandquist (1997) (Correct)
....formulation, the submatrix in question need not have consecutive rows or columns. It is clear that i(n; t) n Gamma z(n; t) Generalizing this problem to s Theta t submatrices of a zero one matrix of order m Theta n leads naturally to the numbers z(m; n; s; t) and i(m; n; s; t) Bollob as [4] has shown that 2ex(n; K(s; t) z(n; n; s; t) where ex(n; F ) denotes the maximum number of edges in a graph on n vertices that does not contain F as a subgraph. In contrast with the classical Tur an numbers, definitive general results are not known in the bipartite case. The initial search ....
....In contrast with the classical Tur an numbers, definitive general results are not known in the bipartite case. The initial search for numerical values of z(n; t) t = 3; 4; 5 : n = 4; 5; 6; due to Zarankiewicz; Sierpinski; Brzezinski; Culik; Guy; and Zn am, is chronicled in [4], as is the history of research (due to Hartman, Mycielski and Ryll Nardzewski; and Rieman) leading to asymptotic bounds on z(n; 2) and on z(m; n; s; t) the latter set of results are due to Kov ari, S os and Tur an; Hylt en Cavallius; and Zn am) The asymptotics of the numbers z(n; n; 2; t) t ....
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B. BOLLOB ' AS, "Extremal Graph Theory," Academic Press, London, 1978.
The Complexity of Identifying Large Equivalence Classes - Binderup, Frandsen.. (1998) (2 citations) (Correct)
....[ 2; n] has its minimum for s = where the value is 24 Theorem 3.3 Let G be an undirected graph with n vertices. For integer k 2, if the number of edges is more than then there is a clique of size k in G. Proof The theorem is an immediate consequence of Tur an s Theorem (see e.g. [4] or [8] Lemma 3.1 Let G be an undirected graph with n vertices and k 2 an integer. If the number of edges is less than 2 (n k 1) then there exists either 3 independent vertices or a clique of size k. Proof Assume no 3 vertices are independent and that G has no clique of size k. Let v be ....
B. Bollobas, Extremal Graph Theory. Academic Press (1978).
Moore Graphs and Beyond: - The (Correct)
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B. Bollobas, Extremal Graph Theory, Academic Press, New York (1978).
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B. Bollob as, "Extremal Graph Theory," Academic Press, London, 1978.
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B. Bollob as, Extremal graph theory. In: R. L. Graham, M. Grotschel and L. Lovasz (ed.), Handbook of Combinatorics, Volume 2, Elsevier, Amsterdam, 1995; pp. 1231-1292.
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B. Bollob as, Extremal Graph Theory, Academic Press, London, 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, 1978.
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B. Bollobas, Extremal Graph Theory, Academic Press, London (1978).
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B. Bollob as, Extremal Graph Theory, LMS Monographs vol. 11, Academic Press, London, New York, San Francisco, 1978,
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A Characterization of Uniquely Vertex Colorable.. - Hajiabolhassan.. (Correct)
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B. Bollob as, Extremal Graph Theory, LMS Monographs vol. 11, Academic Press, London, New York, San Francisco, 1978,
On Graph Problems in a Semi-Streaming Model - Feigenbaum, Kannan, McGregor.. (2004) (1 citation) (Correct)
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B. Bollobas. Extremal Graph Theory. Academic Press, New York, 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 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 1978.
Large Planar Subgraphs in Dense Graphs - Kühn, Osthus, Taraz (2003) (Correct)
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B. Bollobas, Extremal Graph Theory, Academic Press 1978.
A Density Result For Random Sparse Oriented Graphs And Its .. - Donadelli, Kohayakawa (Correct)
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Bela Bollobas, Extremal graph theory, Academic Press Inc. [Harcourt Brace Jovanovich Publishers ], London, 1978.
Szemeredi's Regularity Lemma And Quasi-Randomness - Kohayakawa, Rödl (2002) (Correct)
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Turán numbers of bipartite graphs and related.. - Alon, Krivelevich.. (Correct)
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B. Bollobas, Extremal Graph Theory, Academic Press, London, 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, London, 1978.
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Bollob' as, B., Extremal graph theory, Academic Press, London, 1978.
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B. Bollobas. Extremal Graph Theory. Academic Press, London, 1978.
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