## Books in graphs (2008)

Citations: | 1886 - 20 self |

### BibTeX

@MISC{Bollobás08booksin,

author = {Béla Bollobás and Vladimir Nikiforov},

title = {Books in graphs},

year = {2008}

}

### Years of Citing Articles

### OpenURL

### 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

2076 | An introduction to probability theory and its applications - Feller - 1968 |

1992 | On the evolution of random graphs
- Erdös, Rényi
- 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], [... |

1206 |
Extremal Graph Theory
- Bollob'as
- 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(... |

165 | Modern graph theory, Graduate texts - Bollobás |

160 |
Graph theory and probability
- Erdős
- 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... |

82 |
On colouring random graphs
- Grimmett, McDiarmid
- 1975
(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... |

81 | A method for solving extremal problems in graph theory, In: Theory of Graphs - Simonovits - 1968 |

63 |
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(... |

47 |
Topological cliques in random graphs
- Bollobás, Catlin
- 1981
(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... |

38 | Problems and results in combinatorial analysis and combinatorial number theory - Erdős - 1988 |

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

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 a theorem of Rademacher–Turán - Erdős - 1962 |

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

5 | More from the Good Book - Faudree, Rousseau, et al. - 1978 |

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... |

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

3 | On the book size of graphs with large minimum degree, Studia Sci - Erdős, Faudree, et al. - 1995 |

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

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... |