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On the Evolution of Random Graphs
 PUBLICATION OF THE MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES
, 1960
"... his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_ ..."
Abstract

Cited by 1836 (8 self)
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his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_
Supersaturated graphs and hypergraphs
 Combinatorica
, 1983
"... We shall consider graphs (hypergraphs) without loops and multiple edges. Let Ybe a family of so called prohibited graphs and ex (n, Y) denote the maximum number of edges (hyperedges) a graph (hypergraph) on n vertices can have without containing subgraphs from Y A graph (hypergraph) will be called s ..."
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Cited by 28 (0 self)
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We shall consider graphs (hypergraphs) without loops and multiple edges. Let Ybe a family of so called prohibited graphs and ex (n, Y) denote the maximum number of edges (hyperedges) a graph (hypergraph) on n vertices can have without containing subgraphs from Y A graph (hypergraph) will be called supersaturated if it has more edges than ex (n, Y). If G has n vertices and ex (n, Y)=k edges (hyperedges), then it always contains prohibited subgraphs. The basic question investigated here is: At least how many copies of L E Y must occur in a graph G " on n vertices with ex (n, Y)+k edges (hy peredges)? Notation. In this paper we shall consider only graphs and hypergraphs without loops and multiple edges, and all hypergraphs will be uniform. If G is a graph or hypergraph, e(G), v(G) and y(G) will denote the number of edges, vertices and the chromatic number of G, respectively. The first upper index (without brackets) will denote the number of vertices: G", S", T ",P are graphs of order n. Kph)(m r,..., m p) denotes the huniform hypergraph with m,+...+mp vertices partitioned into classes Cl,..., C p, where JQ=mi (i=1,..., p) and the hyperedges of this graph are those htuples, which have at most one vertex in each C i. For h=2 KP (ni l,..., nt p) is the ordinary complete ppartite graph. In some of our assertions we shall say e.g. that "changing o(nl) edges in G "... ". (Of course, o() cannot be applied to one graph.) As a matter of fact, in such cases we always consider a sequence of graphs G " and n.
Some extremal problems in graph theory
, 1969
"... We consider only graphs without loops and multiple edges. G n denotes a graph of n vertices, v(G) , e(G) and X(G) denote the number of vertices, edges, and the chromatic number of the graph G respectively. The star of a vertex x will be denoted by st x (that is the set of vertices joined to x), the ..."
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Cited by 24 (0 self)
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We consider only graphs without loops and multiple edges. G n denotes a graph of n vertices, v(G) , e(G) and X(G) denote the number of vertices, edges, and the chromatic number of the graph G respectively. The star of a vertex x will be denoted by st x (that is the set of vertices joined to x), the valency of x will be denoted by 6(x), K(m,n) denotes the complete bichromatic graph with m and n vertices in its classes. {KCm,n)r} is the graph obtained from K(m, n) omitting r ( r< _ min (m, n)) independent edges. Thus { K(4,4) 4} = C is the graph formed by the vertices and edges of a cube. a graph G n Let us denote by f ( n; L1 _. L %, ) the maximum number of edges can have if it does not contain any L ~ as a subgraph. abbreviated by f ( n). If it does not cause any confusion, f ( n; L i,..., L), ) will be According to [1] ( 1) f(n; K(Z,m)) = Om(n2 C) (£<_m) exists, and perhaps
On a valence problem in extremal graph theory
 Discrete Math
, 1972
"... Abstract. Let L # Kp be a pchromatic graph and e be an edge of L such that L e is (p 1)chromatic. If G n is a graph of n vertices without containing L but containing Kp, then the minimum valence of G " is n (1p1 ..."
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Cited by 19 (1 self)
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Abstract. Let L # Kp be a pchromatic graph and e be an edge of L such that L e is (p 1)chromatic. If G n is a graph of n vertices without containing L but containing Kp, then the minimum valence of G " is n (1p1
Additive Spanners and (α, β)Spanners
"... An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). How ..."
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Cited by 6 (2 self)
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An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). However no other additive spanners are known to exist. In this paper we develop a couple of new techniques for constructing (α, β)spanners. Our first result is an additive (1, 6)spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively wellapproximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparsenessdistortion tradeoffs. Our second result addresses the problem of which (α, β)spanners can be computed efficiently, ideally in linear time. We show that for any k, a (k, k − 1)spanner with size O(kn 1+1/k) can be found in linear time, and further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
Cubesupersaturated graphs and related problems
 IN PROGRESS IN GRAPH THEORY
, 1982
"... In this paper we shall consider ordinary graphs, that is, graphs without loops and multiple edges. Given a graph L, ex (n, L) will denote the maximum number of edges a graph G " of order n can have without containing any L. Determining ex(n,L), or at least finding good bounds on it will be call ..."
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Cited by 5 (0 self)
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In this paper we shall consider ordinary graphs, that is, graphs without loops and multiple edges. Given a graph L, ex (n, L) will denote the maximum number of edges a graph G " of order n can have without containing any L. Determining ex(n,L), or at least finding good bounds on it will be called TURÁN TYPE EXTREMAL PROBLEM.