## Books in graphs (2008)

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Citations: | 1986 - 20 self |

### BibTeX

@MISC{Bollobás08booksin,

author = {Béla Bollobás and Vladimir Nikiforov},

title = {Books in graphs},

year = {2008}

}

### Years of Citing Articles

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### Abstract

A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least

### Citations

2275 | An Introduction to Probability Theory and its Applications - Feller - 1971 |

2125 | On the evolution of random graphs
- Erdos, Renyi
- 1960
(Show Context)
Citation Context ...er d. About twenty years ago Erdös [7], [8] used random graphs to tackle problems concerning Ramsey numbers and the relationship between the girth and the chromatic number of a graph. Erdös and Rényi =-=[9]-=-, [10] initiated the study of random graphs for their own sake, and proved many beautiful and striking results. The graph invariants investigated in recent years include the clique number [5], [13], [... |

1265 |
Extremal Graph Theory
- Bollobas
- 1978
(Show Context)
Citation Context ... we always use constants following from generous calculations, so the reader should not be surprised if he can see the inequalities with better constants. We shall use the notation and terminology of =-=[1]-=-. We shall denote by Tk(x) the set of vertices at distance k from x: Tk(x)= {y(EG:d(x,y) = k} and write Nk(x) for the set of vertices within distance k: k Nk(x) = U r,(x). 1 = 0 Thus diam G = d if Nd(... |

274 |
Combinatorial problems and exercises
- Lovász
- 1979
(Show Context)
Citation Context ...s with both parts larger than C log n; for some C > 0; independent of n: Hence, Theorem 1 is essentially best possible. - Theorem 1 implies the Erdos-Stone-Bollobás theorem. Indeed, in [9] (see also =-=[11]-=-, Problem 11.8) it is proved that if ks (G) > 0; then (s+ 1) ks+1 (G) sks (G) n s sks (G) (s 1) ks1 (G) n s 1 : Thus, if e (G) (1 1=r + c)n2=2; wesnd that kr+1 (G) > (c=rr)nr+1; and so G c... |

174 |
Modern Graph Theory, Graduate Texts
- Bollobás
- 1998
(Show Context)
Citation Context .... Also, in our setup c may be a function of n; say c = 1= ln lnn and the bounds on g (r; c; n) remain meaningful; such results are beyond the scope of the papers mentioned above. Our notation follows =-=[1]-=-. Thus, Kr (s1; : : : ; sr) denotes a complete r-partite graph with parts of size s1; : : : ; sr; and an r-clique is a complete subgraph on r vertices. We write E (G) for the edge set of a graph G and... |

169 |
Graph Theory and Probability
- Erdos
- 1961
(Show Context)
Citation Context ...fy (log n)/d - 3 log log n -> oo, 2rf_Imd'/'nd+x - log n -» oo and dd~2md~l/nd — log n -» -oo then almost every graph with n labelled vertices and m edges has diameter d. About twenty years ago Erdös =-=[7]-=-, [8] used random graphs to tackle problems concerning Ramsey numbers and the relationship between the girth and the chromatic number of a graph. Erdös and Rényi [9], [10] initiated the study of rando... |

147 | On a problem of K - Kövári, Sós, et al. - 1954 |

146 | On the structure of linear graphs
- Erdős, Stone
- 1946
(Show Context)
Citation Context ...os-Stone theorem Main results This note is part of an ongoing project aiming to renovate some classical results in extremal graph theory, see, e.g., [5], [12, 15]. Recall a result of Erdos and Stone =-=[7]-=-: every graph with n vertices and dcn2e edges contains a complete bipartite subgraph with each part of size ba log nc, where c; a > 0 are independent of n: One generalization of this result stems from... |

86 | A method for solving extremal problems in graph theory, stability problems - Simonovits - 1966 |

85 |
On colouring random graphs
- Grimmett, klcDiarmid
(Show Context)
Citation Context ...nyi [9], [10] initiated the study of random graphs for their own sake, and proved many beautiful and striking results. The graph invariants investigated in recent years include the clique number [5], =-=[13]-=-, [17], the chromatic number [5], [13], the edge chromatic number [11], the circumference [16], [19], and the degree sequence [4]. The aim of this paper is to give rather precise results concerning th... |

67 |
A Course in Probability Theory (2nd ed
- Chung
- 1974
(Show Context)
Citation Context ...air of vertices x,y we have and |W,_,(*) To justify (5) note that for \Nd_2(x) u Nd_2(y)\ < 4Í»*-2, (4) |r,_,(*) n Td_x(y)\ < "i" «,2o«)"-1-* < IpV-W-i (5) * = i n ^-,00)1 < 2pn\Nd_x(x) n ivrf_,(^)|. =-=(6)-=- |//2j 2 2 m,(/-«)¿-|-'<3»),W'í"2<A2n2J-3> *=i and c, —» oo as n —* oo. Furthermore, 2 2 ^(/m)rf-1-*<3w(/_1 = 6>2''-2n2<'-3, *-[(«/+1)/2-] so (5) does hold. Relations (2)-(6) imply and |^_,(x) u Nd_2(... |

51 |
Cliques in random graphs
- Bollobás, Erdős
- 1976
(Show Context)
Citation Context ...nd Rényi [9], [10] initiated the study of random graphs for their own sake, and proved many beautiful and striking results. The graph invariants investigated in recent years include the clique number =-=[5]-=-, [13], [17], the chromatic number [5], [13], the edge chromatic number [11], the circumference [16], [19], and the degree sequence [4]. The aim of this paper is to give rather precise results concern... |

41 | Problems and results in combinatorial analysis - ERDÖS - 1976 |

38 | On the connection between chromatic number, maximal clique and minimal degree of a graph - Andrásfai, Erdős, et al. - 1974 |

19 | On the structure of edge graphs
- Bollobás, Erdos
- 1973
(Show Context)
Citation Context ...complete (r + 1)-partite graph with each part of size g (r; c; n) ; where, for r and csxed, g (r; c; n) tends to in nity with n. In [3] Bollobás and Erdos found that g (r; c; n) = (log n) ; and in =-=[2]-=-, [4], [6], and [8] the function g (r; c; n) was determined with great precision. In this note we deduce the Erdos-Stone-Bollobás result from weaker premises. Also, in our setup c may be a function o... |

14 | On a theorem of Rademacher-Turán - Erdős - 1962 |

14 | Some of my favourite problems in various branches of combinatorics - Erdős - 1992 |

14 |
Diameters of random graphs
- Klee, Larmann
- 1981
(Show Context)
Citation Context ...2 by Moon and Moser [18], the case diam G < oo by Erdös and Rényi [9], and the diameter of components of sparse graphs by Korshunov [15]. When I was writing this paper, I learned that Klee and Larman =-=[14]-=- proved some results concerning the case diam G = d for fixed values of d. The main result of Klee and Larman [14] is that if d > 3 is a fixed natural number and m = m(n) satisfies md/nd+x - log« -» o... |

14 |
On the complete subgraph of a random graph
- Matula
- 1970
(Show Context)
Citation Context ...], [10] initiated the study of random graphs for their own sake, and proved many beautiful and striking results. The graph invariants investigated in recent years include the clique number [5], [13], =-=[17]-=-, the chromatic number [5], [13], the edge chromatic number [11], the circumference [16], [19], and the degree sequence [4]. The aim of this paper is to give rather precise results concerning the diam... |

13 | On the Erdos-Stone theorem
- Chvátal, Szemerédi
- 1981
(Show Context)
Citation Context ...r + 1)-partite graph with each part of size g (r; c; n) ; where, for r and csxed, g (r; c; n) tends to in nity with n. In [3] Bollobás and Erdos found that g (r; c; n) = (log n) ; and in [2], [4], =-=[6]-=-, and [8] the function g (r; c; n) was determined with great precision. In this note we deduce the Erdos-Stone-Bollobás result from weaker premises. Also, in our setup c may be a function of n; say c... |

9 | On Ramsey numbers for books - Rousseau, Sheehan - 1978 |

8 |
Proof of a conjecture of Bollobás and Kohayakawa on the Erdos-Stone theorem
- Ishigami
(Show Context)
Citation Context ...rtite graph with each part of size g (r; c; n) ; where, for r and csxed, g (r; c; n) tends to in nity with n. In [3] Bollobás and Erdos found that g (r; c; n) = (log n) ; and in [2], [4], [6], and =-=[8]-=- the function g (r; c; n) was determined with great precision. In this note we deduce the Erdos-Stone-Bollobás result from weaker premises. Also, in our setup c may be a function of n; say c = 1= ln ... |

6 | More from the good book - Faudree, Rousseau, et al. - 1978 |

6 |
The Nordhaus-Stewart-Moon-Moser inequality
- Khadµziivanov, Nikiforov
- 1978
(Show Context)
Citation Context ...rtite subgraphs with both parts larger than C log n; for some C > 0; independent of n: Hence, Theorem 1 is essentially best possible. - Theorem 1 implies the Erdos-Stone-Bollobás theorem. Indeed, in =-=[9]-=- (see also [11], Problem 11.8) it is proved that if ks (G) > 0; then (s+ 1) ks+1 (G) sks (G) n s sks (G) (s 1) ks1 (G) n s 1 : Thus, if e (G) (1 1=r + c)n2=2; wesnd that kr+1 (G) > (c=rr)n... |

5 |
Turans theorem inverted, submitted for publication. Preprint available at http://arxiv.org/abs/0707.3394
- Nikiforov
(Show Context)
Citation Context ...ique; number of cliques; r-partite graph; Erdos-Stone theorem Main results This note is part of an ongoing project aiming to renovate some classical results in extremal graph theory, see, e.g., [5], =-=[12, 15]-=-. Recall a result of Erdos and Stone [7]: every graph with n vertices and dcn2e edges contains a complete bipartite subgraph with each part of size ba log nc, where c; a > 0 are independent of n: One... |

5 | Stability for large forbidden graphs, submitted for publication. Preprint available at http://arxiv.org/abs/0707.2563 - Nikiforov |

5 | Graphs with many copies of a given subgraph, submitted for publication. Preprint available at http://arxiv:0711.3493 - Nikiforov |

5 | Complete r-partite subgraphs of dense r-graphs, submitted for publication. Preprint available at http://arxiv.org/abs/0711.1185
- Nikiforov
(Show Context)
Citation Context ...ique; number of cliques; r-partite graph; Erdos-Stone theorem Main results This note is part of an ongoing project aiming to renovate some classical results in extremal graph theory, see, e.g., [5], =-=[12, 15]-=-. Recall a result of Erdos and Stone [7]: every graph with n vertices and dcn2e edges contains a complete bipartite subgraph with each part of size ba log nc, where c; a > 0 are independent of n: One... |

4 | On the Book Size of Graphs with Large Minimum Degree, Studia - Erdos, Faudree, et al. - 1995 |

4 | A note on Ramsey numbers for books, submitted - Nikiforov, Rousseau |

4 |
On the chromatic index of almost all graphs
- Erdös, Wilson
- 1977
(Show Context)
Citation Context ... and proved many beautiful and striking results. The graph invariants investigated in recent years include the clique number [5], [13], [17], the chromatic number [5], [13], the edge chromatic number =-=[11]-=-, the circumference [16], [19], and the degree sequence [4]. The aim of this paper is to give rather precise results concerning the diameter. Recall that the diameter diam G of a connected graph is th... |

4 |
An extension of the Erdos-Stone theorem
- Bollobás, Kohayakawa
- 1994
(Show Context)
Citation Context ...ete (r + 1)-partite graph with each part of size g (r; c; n) ; where, for r and csxed, g (r; c; n) tends to in nity with n. In [3] Bollobás and Erdos found that g (r; c; n) = (log n) ; and in [2], =-=[4]-=-, [6], and [8] the function g (r; c; n) was determined with great precision. In this note we deduce the Erdos-Stone-Bollobás result from weaker premises. Also, in our setup c may be a function of n; ... |

3 | A lower bound for the largest number of triangles with a common edge, manuscript - Edwards |

2 | Solution of a problem of P. Erdős about the maximum number of triangles with a common edge in a graph - Khadˇziivanov, Nikiforov - 1979 |

2 |
theory, an introductory course, Graduate Texts in
- Graph
- 1979
(Show Context)
Citation Context ...= {y(EG:d(x,y) = k} and write Nk(x) for the set of vertices within distance k: k Nk(x) = U r,(x). 1 = 0 Thus diam G = d if Nd(x) = V(G) for every vertex x and Nd_x(y) =£ V(G) for some vertex y. As in =-=[3]-=- we write %(n, /'(edge) = p) for the discrete probability space consisting of the 2® labelled graphs of order n in which the probability of a fixed graph with m edges ispm(l - p)®~m. Equivalently, in ... |

2 |
Almost all (0; 1) matrices are primitive
- Moon, Moser
- 1966
(Show Context)
Citation Context ...the maximum of the distances between vertices, and a disconnected graph has infinite diameter. The diameter of a random graph has hardly been studied, apart from the case diam G = 2 by Moon and Moser =-=[18]-=-, the case diam G < oo by Erdös and Rényi [9], and the diameter of components of sparse graphs by Korshunov [15]. When I was writing this paper, I learned that Klee and Larman [14] proved some results... |

1 |
girth and maximum degree
- number
- 1978
(Show Context)
Citation Context ...NDOM GRAPHS 47 with probability at least 1 - n~K~x, Lemma 4 gives that every pair of vertices x,y satisfies Note that \T*(x,y)\<mk. (1) and *d-.<*) n Nd_x(y) C Nd_2(x) u Nd_2(y) C (Td_x(x) n Td_x(y)) =-=(2)-=- rf-i Td_x(x) n r^.o-) c U Tä.l.k(Xt(x,y)). (3) * = 1 From Lemma 3 and inequality (3) we find that with probability at least 1 — 2« ~K~ ' for every pair of vertices x,y we have and |W,_,(*) To justify... |

1 |
sequences of random graphs
- Degree
- 1981
(Show Context)
Citation Context ...nvariants investigated in recent years include the clique number [5], [13], [17], the chromatic number [5], [13], the edge chromatic number [11], the circumference [16], [19], and the degree sequence =-=[4]-=-. The aim of this paper is to give rather precise results concerning the diameter. Recall that the diameter diam G of a connected graph is the maximum of the distances between vertices, and a disconne... |

1 |
On the diameter of graphs
- Korshunov
- 1971
(Show Context)
Citation Context ... random graph has hardly been studied, apart from the case diam G = 2 by Moon and Moser [18], the case diam G < oo by Erdös and Rényi [9], and the diameter of components of sparse graphs by Korshunov =-=[15]-=-. When I was writing this paper, I learned that Klee and Larman [14] proved some results concerning the case diam G = d for fixed values of d. The main result of Klee and Larman [14] is that if d > 3 ... |

1 |
Hamiltonian cycles in random graphs, Discrete Math. 14
- Posa
- 1976
(Show Context)
Citation Context ... striking results. The graph invariants investigated in recent years include the clique number [5], [13], [17], the chromatic number [5], [13], the edge chromatic number [11], the circumference [16], =-=[19]-=-, and the degree sequence [4]. The aim of this paper is to give rather precise results concerning the diameter. Recall that the diameter diam G of a connected graph is the maximum of the distances bet... |