Results 1  10
of
17
The number of cliques in graphs of given order and size
, 2008
"... Let kr (n,m) denote the minimum number of rcliques in graphs with n vertices and m edges. For r = 3,4 we give a lower bound on kr (n,m) that approximates kr (n,m) with an error smaller than n r / ( n 2 − 2m). The solution is based on a constraint minimization of certain multilinear forms. Our proo ..."
Abstract

Cited by 38 (1 self)
 Add to MetaCart
(Show Context)
Let kr (n,m) denote the minimum number of rcliques in graphs with n vertices and m edges. For r = 3,4 we give a lower bound on kr (n,m) that approximates kr (n,m) with an error smaller than n r / ( n 2 − 2m). The solution is based on a constraint minimization of certain multilinear forms. Our proof combines a combinatorial strategy with extensive analytical arguments.
Asymptotic structure of graphs with the minimum number of triangles, arXiv:1203.4393
"... We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density. 1 ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density. 1
Extremal Graphs With Bounded Densities Of Small Subgraphs
, 1998
"... Let Ex(n, k, ) denote the maximum number of edges of an nvertex graph in which every subgraph of k vertices has at most edges. Here we summarize some known results of the problem of determining Ex(n, k, ), give simple proofs, and find some new estimates and extremal graphs. Besides proving new res ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
Let Ex(n, k, ) denote the maximum number of edges of an nvertex graph in which every subgraph of k vertices has at most edges. Here we summarize some known results of the problem of determining Ex(n, k, ), give simple proofs, and find some new estimates and extremal graphs. Besides proving new results, one of our main aims is to show how the classical Turan theory can be applied to such problems. The case = k 2  1 is the famous result of Turan. Address correspondence to Professor Griggs. Email address: griggs@math.sc.edu 1 Research supported in part by grants NSA/MSP/MDA90492H3053 and 95H1024 and by NSF DMS 9701211. 2 Research supported in part by Hungarian Research Grant OTKA 1909. 1 Extremal graphs June 8, 1998 0. INTRODUCTION AND NOTATION We consider undirected graphs G without loops and multiple edges. The set of vertices, the set of edges, and the chromatic number are denoted by V (G), E(G), and #(G), respectively. We denote the number of vertices (resp....
Counting substructures I: color critical graphs
, 2009
"... Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits [10], who proved that there is one copy of F, and of Rademacher, E ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits [10], who proved that there is one copy of F, and of Rademacher, Erdős [1, 2] and LovászSimonovits [4], who proved similar counting results when F is a complete graph. One of the simplest cases of our theorem is the following new result. There is an absolute positive constant c such that if n is sufficiently large and 1 ≤ q < cn, then every n vertex graph with ⌊n2 /4 ⌋ + q edges contains at least n
Counting substructures III: quadruple systems
, 2009
"... For various quadruple systems F, we give asymptotically sharp lower bounds on the number of copies of F in a quadruple system with a prescribed number of vertices and edges. Our results extend those of Füredi, Keevash, Pikhurko, Simonovits and Sudakov who proved under the same conditions that there ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
For various quadruple systems F, we give asymptotically sharp lower bounds on the number of copies of F in a quadruple system with a prescribed number of vertices and edges. Our results extend those of Füredi, Keevash, Pikhurko, Simonovits and Sudakov who proved under the same conditions that there is one copy of F. Our proofs use the hypergraph removal Lemma and stability results for the corresponding Turán problem proved by the above authors. 1
Books versus Triangles
, 2009
"... A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ⌊n 2 /4 ⌋ + 1 edges. Rademacher proved that G contains at least ⌊n/2 ⌋ triangles, and Erdős conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following “line ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ⌊n 2 /4 ⌋ + 1 edges. Rademacher proved that G contains at least ⌊n/2 ⌋ triangles, and Erdős conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following “linear combination ” of these two results. Suppose that α ∈ (1/2, 1) and the maximum size of a book in G is less than αn/2. Then G contains at least α(1 − α) n2 4 − o(n2) triangles as n → ∞. This is asymptotically sharp. On the other hand, for every α ∈ (1/3, 1/2), there exists β> 0 such that G contains at least βn 3 triangles. It remains an open problem to determine the largest possible β in terms of α. Our proof uses the RuzsaSzemerédi theorem. 1
Counting substructures II: triple systems
, 2009
"... For various triple systems F, we give tight lower bounds on the number of copies of F in a triple system with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of various authors who proved that there is one copy of F. A sample r ..."
Abstract
 Add to MetaCart
(Show Context)
For various triple systems F, we give tight lower bounds on the number of copies of F in a triple system with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of various authors who proved that there is one copy of F. A sample result is the following: FürediSimonovits [10] and independently KeevashSudakov [15] settled an old conjecture of Sós [28] by proving that the maximum number of triples in an n vertex triple system (for n sufficiently large) that contains no copy of ⌊n/2⌋
On the minimal density of . . .
, 2008
"... For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that (t − 1) t − 2 g3(ρ) = t(t − ρ(t + 1)) t + � �2 t(t − ρ(t + 1)) t2 (t + 1) 2 where t def ..."
Abstract
 Add to MetaCart
For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that (t − 1) t − 2 g3(ρ) = t(t − ρ(t + 1)) t + � �2 t(t − ρ(t + 1)) t2 (t + 1) 2 where t def
Two questions of Erdős on hypergraphs above the Turán
, 2011
"... For ordinary graphs it is known that any graph G with more edges than the Turán number of Ks must contain several copies of Ks, and a copy of K−s+1, the complete graph on s+ 1 vertices with one missing edge. Erdős asked if the same result is true for K3s, the complete 3uniform hypergraph on s ver ..."
Abstract
 Add to MetaCart
(Show Context)
For ordinary graphs it is known that any graph G with more edges than the Turán number of Ks must contain several copies of Ks, and a copy of K−s+1, the complete graph on s+ 1 vertices with one missing edge. Erdős asked if the same result is true for K3s, the complete 3uniform hypergraph on s vertices. In this note we show that for small values of n, the number of vertices in G, the answer is negative for s = 4. For the second property, that of containing a K3s+1, we show that for s = 4 the answer is negative for all large n as well, by proving that the Turán density of