Results 1  10
of
40
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 ..."
Abstract

Cited by 38 (0 self)
 Add to MetaCart
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.
The early evolution of the Hfree process
, 2009
"... The Hfree process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal Hfree graph obtained at t ..."
Abstract

Cited by 33 (4 self)
 Add to MetaCart
(Show Context)
The Hfree process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal Hfree graph obtained at the end of the process. When H is strictly 2balanced, we show that for some c> 0, with high probability as n → ∞, the minimum degree in G is at least cn 1−(vH −2)/(eH−1) (log n) 1/(eH −1). This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs Kr,r with r ≥ 5. When H is a complete graph Ks with s ≥ 5 we show that for some C> 0, with high probability the independence number of G is at most Cn 2/(s+1) (log n) 1−1/(eH −1). This gives new lower bounds for Ramsey numbers R(s, t) for fixed s ≥ 5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.
Problems and results in extremal combinatorics  II
 DISCRETE MATHEMATICS
, 2003
"... Extremal Combinatorics is one of the central areas in Discrete Mathematics. It deals with problems that are often motivated by questions arising in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, inc ..."
Abstract

Cited by 21 (0 self)
 Add to MetaCart
Extremal Combinatorics is one of the central areas in Discrete Mathematics. It deals with problems that are often motivated by questions arising in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics considered here include questions in Extremal Graph Theory, Polyhedral Combinatorics and Probabilistic Combinatorics. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably biased, and as the title of the paper suggests, it is a sequel to a previous paper [2] of the same flavour, and hopefully a predecessor of another related future paper. Each section of this paper is essentially self contained, and can be read separately.
Simultaneously Satisfying Linear Equations Over F2: MaxLin2 and MaxrLin2 Parameterized Above Average
 IN FSTTCS 2011, LIPICS
, 2011
"... In the parameterized problem MAXLIN2AA[k], we are given a system with variables x1,..., xn consisting of equations of the form ∏i∈I x i = b, where x i, b ∈ {−1, 1} and I ⊆ [n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equa ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
(Show Context)
In the parameterized problem MAXLIN2AA[k], we are given a system with variables x1,..., xn consisting of equations of the form ∏i∈I x i = b, where x i, b ∈ {−1, 1} and I ⊆ [n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2 + k, where W is the total weight of all equations and k is the parameter (if k = 0, the possibility is assured). We show that MAXLIN2AA[k] has a kernel with at most O(k 2 log k) variables and can be solved in time 2 O(k log k) (nm) O(1). This solves an open problem of Mahajan et al. (2006). The problem MAXrLIN2AA[k, r] is the same as MAXLIN2AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on MAXrLIN2AA[k, r] which implies that MAXrLIN2AA[k, r] has a kernel with at most (2k − 1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f: {−1, 1} n → R whose Fourier expansion (which is a multilinear polynomial) is of degree r. We show applicability of the lower bound by giving a new proof of the EdwardsErdős bound (each connected graph on n vertices and m edges has a bipartite subgraph with at least m/2 + (n − 1)/4 edges) and obtaining a generalization.
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 ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
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.
MaxCut Parameterized Above the EdwardsErdős Bound
 IN ICALP 2012, PART I, LECT. NOTES COMPUT. SCI. 7391
, 2012
"... ..."
The Size of the Largest Bipartite Subgraphs
, 1998
"... Simple proofs are given for results of Edwards concerning the size of the largest bipartite subgraphs of a graph. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
Simple proofs are given for results of Edwards concerning the size of the largest bipartite subgraphs of a graph.
Precisely Answering Multidimensional Range Queries Without Privacy Breaches
 ESORICS
, 2003
"... This paper investigates the privacy breaches caused by multidimensional range (MDR) sum queries in OLAP systems. We show that existing inference control methods are generally ineffective or infeasible for MDR queries. We then consider restricting users to even MDR queries (that is, the MDR queries ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
This paper investigates the privacy breaches caused by multidimensional range (MDR) sum queries in OLAP systems. We show that existing inference control methods are generally ineffective or infeasible for MDR queries. We then consider restricting users to even MDR queries (that is, the MDR queries involving even number of data values). We show that the collection of such even MDR queries is safe if and only if a special set of sumtwo queries (that is, queries involving exactly two values) is safe. On the basis of this result, we give an efficient method to decide the safety of even MDR queries. Besides safe even MDR queries we show that any odd MDR query is unsafe. Moreover, any such odd MDR query is different from the union of some even MDR queries by only one tuple. We also extend those results to the safe subsets of unsafe even MDR queries.