Results 1  10
of
13
Books in graphs
, 2008
"... A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) ..."
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Cited by 1821 (20 self)
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A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least
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 ..."
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Cited by 12 (0 self)
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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.
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 ..."
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Cited by 8 (1 self)
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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....
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 ..."
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Cited by 4 (0 self)
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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
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 ..."
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Cited by 2 (2 self)
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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
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 ..."
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Cited by 2 (2 self)
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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
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 ..."
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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
2 I a unique G(n; I) which contains no triangle [3]. Rademacher
, 1963
"... Let G(n;m) denote a graph of n vertices and m edges. Vertices of G will be denoted by x 1,...,y 1...; edges will be denoted by (x, y) and triangles by (x, y, z). (G x 1 x 2 xk) will denote the graph G from which the vertices x 1,..., xk and all edges incident to them have been omitted. G (x,,x,) ..."
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Let G(n;m) denote a graph of n vertices and m edges. Vertices of G will be denoted by x 1,...,y 1...; edges will be denoted by (x, y) and triangles by (x, y, z). (G x 1 x 2 xk) will denote the graph G from which the vertices x 1,..., xk and all edges incident to them have been omitted. G (x,,x,) denotes the graph G from which the edge (x i,x) has been omitted. A special case of a well known theorem of Turin states 2 that every G(n; + 1) contains a triangle and that there is
On qanalogues and stability theorems
"... Abstract. In this survey recent results about qanalogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the qanalogues of Kneser graphs. For the proof one needs results on the number of 0secant subspaces of point sets, so i ..."
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Abstract. In this survey recent results about qanalogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the qanalogues of Kneser graphs. For the proof one needs results on the number of 0secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.