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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_ ..."
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Cited by 2043 (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 32 (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 &quot; 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&quot;, S&quot;, T &quot;,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 &quot;changing o(nl) edges in G &quot;... &quot;. (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 &quot; 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 31 (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 ..."
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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 &quot; 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.
Topology of 2dimensional complexes
"... ABSTRACT. We present results concerning geometric topology of 2dimensional polyhedra, with emphasis on those related to the problem of embeddability into 3manifolds (and thickenings), the problem of resolving arbitrary 2polyhedra by fake surfaces (including an application to reduce the classical ..."
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Cited by 1 (1 self)
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ABSTRACT. We present results concerning geometric topology of 2dimensional polyhedra, with emphasis on those related to the problem of embeddability into 3manifolds (and thickenings), the problem of resolving arbitrary 2polyhedra by fake surfaces (including an application to reduce the classical Whitehead asphericity conjecture to special polyhedra) and existence of nonhomeomorphic 3manifolds with equivalent spines. We also consider different regular neighborhoods of codimension 2 embeddings of polyhedra into manifolds of any dimension. A special section is devoted to algebraic topology of 2polyhedra, cohomology of groups and universal covers. 1. ON EMBEDDABILITY OF 2POLYHEDRA INTO 3MANIFOLDS In this section, we present the basic results on embeddability of 2polyhedra into 3manifolds. We omit 2coefficients from the notation of (co)homology groups. In our notation and terminology, we follow [43]. Throughout this paper we shall work in the PL category. By [4] the same results hold in the topological category. A vertex of a graph is hanging if its degree is one. An edge of a graph is hanging if one of its endpoints is hanging. A link of a point of X is its link in some sufficiently small triangulation of X. 1.1. Fake surfaces and special 2polyhedra. A finite 2polyhedron Q is called a fake surface if each of its points has a neighborhood homeomorphic to one of the following: D2, the book with three pages (T I), or the cone over the complete graph with four vertices (or over the 1skeleton of the 3simplex). See Figure 1.1 for an illustration of these three types of neighborhoods. We will refer to points in fake surfaces as points of type 1, 2 and 3, respectively depending on which of the above three neighborhoods they have. Soap films in 3 exhibit singularities precisely of types 2 and 3. The notion of soap films from differential geometry has proved to be an important tool and object of investigation in algebraic and geometric topology.
DEDICATED TO THE MEMORY OF PAUL TUR.kN ON THE OCCASION OF THE 10TH ANNIVERSARY OF HIS DEATH
, 1986
"... Let G be a graph with vertex set V(G) and edge set E(G), respectively. The set of vertices adjacent to an x e V(G) is denoted by F(x), and the degree of x is d(x) = I F(x)l. For any subset V 'g V(G), let G [ V] denote the subgraph of G induced by the vertices of V '. Further, let K n stan ..."
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Let G be a graph with vertex set V(G) and edge set E(G), respectively. The set of vertices adjacent to an x e V(G) is denoted by F(x), and the degree of x is d(x) = I F(x)l. For any subset V 'g V(G), let G [ V] denote the subgraph of G induced by the vertices of V '. Further, let K n stand for the complete graph on n vertices. It is easily seen (e.g., Erdös [7]) that every graph G with n vertices and in edges contains a bipartite subgraph H such that iE(H)l>IE(G)1/2 = m/2, i.e., every graph can be made bipartite by the omission of at most half of its edges. Erdös and Lovász proved that if G has no triangle, then it can be made bipartite by the omission of m/2 _ M 2/3 (log m) ` edges. On the other hand, Erdös [9] showed by the probability method that for every r, there is a graph G with no cycle of length less than r which cannot be made bipartite by the omission of fewer than in edges. The best exponent in m ' is not known even for r=3, but s. approaches 0 as r becomes large. However, the graphs constructed in [9] are &quot;sparse &quot; (i.e., in = 0(n')), and the aim of this paper is to show that much stronger results can be obtained if we assume that our graph G is not sparse. We will restrict our attention to families of graphs not containing some socalled,lbrbidden subgraph E (Such graphs are also said to be Ffree.) In particular, for trianglefree graphs, i.e., when F = K3, we will prove the following.