## Sparse quasi-random graphs (1989)

Venue: | Combinatorica |

Citations: | 49 - 2 self |

### BibTeX

@ARTICLE{Chung89sparsequasi-random,

author = {Fan Chung and Ronald Graham},

title = {Sparse quasi-random graphs},

journal = {Combinatorica},

year = {1989},

volume = {22},

pages = {217--244}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

138 | On a problem of graph theory
- Erdős, Rényi, et al.
- 1966
(Show Context)
Citation Context ... � (e(y, X) − p|X|) 2 y = o(p 2 n 4 ). which implies DISC(1). � 12 (2)sExample 4: CIRCUIT(2t) �⇒ CYCLE(2t) Proof: There exists a graph on n vertices and (1/2+o(1))n3/2 edges that contains no 4-cycles =-=[15]-=-, but the number of 4-circuits for such a graph is (1 + o(1))n2 . � At present, we do not know the answer to the following: Question: Is it true that “ DISC ⇒ EIG”? 6. Quasi-random classes It is clear... |

88 |
Quasi-random graphs, Combinatorica 9
- Chung, Graham, et al.
- 1989
(Show Context)
Citation Context ...family G1/2 of graphs which satisfies any one of the properties must in fact satisfy all of them. Nevertheless, these and a variety of other properties have now been shown to be equivalent (e.g., see =-=[10, 12]-=-). We call families of graphs which satisfy this equivalence class of properties quasi-random. In particular, as a consequence, a number of rather simple ways are now available for explicitly construc... |

44 | Regularity Lemmas for hypergraphs and quasi-randomness, Random Structures and Algorithms 2 - Chung - 1991 |

44 |
Pseudo-random graphs
- Thomason
- 1987
(Show Context)
Citation Context ...ivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see [5]-[14], [16], [23]-=-=[27]-=-). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn2 edges for some c>0asn→∞. In this paper, we extend our study of quasi-randomness to sparse graphs, i.e., ... |

42 |
graphs, strongly regular graphs and pseudo-random graphs
- Thomason
- 1987
(Show Context)
Citation Context ...h are much sparser, e.g., having e(G) =o(n2 ) edges. We point out that there are some results already available in the literature which have this flavor, most notably, the seminal results of Thomason =-=[26, 27, 28]-=- on (p, α)-jumbled graphs. In these graphs, however, rather more stringent hypotheses are required on the discrepancy of the 2sbehavior of parameters under investigation in order to reach the desired ... |

34 |
Quasi-random set systems
- Chung, Graham
- 1991
(Show Context)
Citation Context ...family G1/2 of graphs which satisfies any one of the properties must in fact satisfy all of them. Nevertheless, these and a variety of other properties have now been shown to be equivalent (e.g., see =-=[10, 12]-=-). We call families of graphs which satisfy this equivalence class of properties quasi-random. In particular, as a consequence, a number of rather simple ways are now available for explicitly construc... |

31 | Constructive bounds for a Ramsey-type problem
- Alon, Krivelevich
- 1997
(Show Context)
Citation Context ...tive. Similarly, #{H <G} := |{ρ : V (H) → V (G) such that x ∼ y in H if and only if ρ(x) ∼ ρ(y) inG}| will denote the number of occurrences of H as an induced subgraph of G. For a function p = p(n) ∈ =-=[0, 1]-=-, we let Gp(n) denotearandom graph with edge probability p. This actually denotes a distribution on the set of graphs on n vertices, which we will usually take to be the set [n] :={1, 2,... ,n}, in wh... |

31 |
An estimate for character sums
- Katz
- 1989
(Show Context)
Citation Context ...e the vertex set V (CP,t) tobe{1, 2,... ,Pt − 1}. Fora, b ∈ V (CP,t), {a, b} is an edge of CP,t provided a + b ∈ G where G = {λ(y) :y ∈ coset g + GF (P )}. It follows by an eigenvalue estimate of Katz=-=[18]-=- that |λ2| ≤(t − 1) √ P for the graph CP,t. SinceCP,t is regular of degree P ,then the family {CP,t} satisfies EIG(2t) fortfixed as P →∞. To see that {CP,t} satisfies W(t), we consider et(u, S) for a ... |

24 |
On universality of graphs with uniformly distributed edges
- Rodl
(Show Context)
Citation Context ...e equivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see [5]-[14], [16], =-=[23]-=--[27]). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn2 edges for some c>0asn→∞. In this paper, we extend our study of quasi-randomness to sparse graphs, i... |

22 |
Quasi-random classes of hypergraphs, Random Structures and Algorithms 1
- Chung
- 1990
(Show Context)
Citation Context ...ties form a large equivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see =-=[5]-=--[14], [16], [23]-[27]). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn2 edges for some c>0asn→∞. In this paper, we extend our study of quasi-randomness to... |

22 | Quasi-random subsets of Zn - Chung, Graham - 1992 |

21 |
The number of submatrices of a given type in a Hadamard matrix and related results
- Frankl, Rödl, et al.
- 1988
(Show Context)
Citation Context ...a large equivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see [5]-[14], =-=[16]-=-, [23]-[27]). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn2 edges for some c>0asn→∞. In this paper, we extend our study of quasi-randomness to sparse gra... |

19 | Szemerédi’s partition and quasirandomness, Random Structures and Algorithms 2 - Simonovits, Sós - 1991 |

16 | Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs, Combinatorica 17 - Simonovits, Sós - 1997 |

14 | Quasi-random tournaments - Chung, Graham - 1991 |

11 |
Non-Ramsey graphs are c log n-universal
- Prömel, Rödl
- 1999
(Show Context)
Citation Context ...of most of these ideas to k-uniform hypergraphs [5, 10]. What are the corresponding extensions in the sparse case? In particular, is there a hypergraph version of the recent result of Prömel and Rödl =-=[22]-=- which shows that “non-Ramsey” graphs are universal? Finally, as mentioned earlier, does DISC imply EIG for every family in Gp? These are just a few of the many intriguing questions which remain. We h... |

9 | On graphs not containing prescribed induced subgraphs, in: A tribute to Paul Erdős - Chung, Graham - 1990 |

6 | Non-averaging subsets and non-vanishing transversals
- Alon, Ruzsa
- 1999
(Show Context)
Citation Context ...was first proved in [3] although it has been independently rediscovered by quite a few people [19, 21]. (Even the case of t = 3 is an interesting exercise.) An elementary proof of (2) can be found in =-=[2]-=-. Here we will show that DISC(2) implies DISC(1). Fact 11: DISC(2) ⇒ DISC(1). Proof: For any subset X of the vertex set V of G, wehave e2(X, X) = � x,x ′ � e(x, y)e(y, x ∈X y ′ ) = � (e(y, X)) 2 . Fro... |

5 |
Cohomological aspects of hypergraphs
- Chung, Graham
- 1992
(Show Context)
Citation Context ... form a large equivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see [5]-=-=[14]-=-, [16], [23]-[27]). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn2 edges for some c>0asn→∞. In this paper, we extend our study of quasi-randomness to spar... |

5 | Dense expanders and pseudo-random bipartite graphs”, Graph Theory and Combinatorics - Thomason - 1988 |

4 | Maximum Cuts and Quasi-random Graphs, Random Graphs 2 (A. Frieze and T. ÃLuczak, eds - Chung, Graham - 1992 |

3 |
private communication
- Gowers
(Show Context)
Citation Context ...tex v of G by m copies of v, denoted by vi, i =1,... ,m, where we assume that ui is adjacent to vj if and only if u and v are adjacent in G. (This construction was first suggested to us by Tim Gowers =-=[17]-=-.) Suppose G [m] has eigenvalues ˜ λ1 ≥| ˜ λ2| ≥.... We will show the following: Fact 10: Let G and G [m] be defined as above with eigenvalues λi and ˜ λi, respectively. Then, (i) ˜ λ1 = mλ1, sup i�=1... |

3 |
Inequalities in quadratic forms
- London
- 1966
(Show Context)
Citation Context ...lies DISC (t) fort≥2. It is known that et(V,V ) ≥ n(pn) t for any graph with average degree pn. This fact was first proved in [3] although it has been independently rediscovered by quite a few people =-=[19, 21]-=-. (Even the case of t = 3 is an interesting exercise.) An elementary proof of (2) can be found in [2]. Here we will show that DISC(2) implies DISC(1). Fact 11: DISC(2) ⇒ DISC(1). Proof: For any subset... |

2 |
An inequality arising in genetical theory, amer
- Mulholland, Smith
- 1969
(Show Context)
Citation Context ...lies DISC (t) fort≥2. It is known that et(V,V ) ≥ n(pn) t for any graph with average degree pn. This fact was first proved in [3] although it has been independently rediscovered by quite a few people =-=[19, 21]-=-. (Even the case of t = 3 is an interesting exercise.) An elementary proof of (2) can be found in [2]. Here we will show that DISC(2) implies DISC(1). Fact 11: DISC(2) ⇒ DISC(1). Proof: For any subset... |