Results 1  10
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11
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) n 2 wi ..."
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Cited by 1802 (19 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....
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
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 ..."
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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
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
Hungarian Academy of Sciences
"... A connected graph G is said to be Fgood if the Ramsey number r(F,G) has the value ( y(F)1) (p(G)1) + s(F), where s(F) is the minimum number of vertices in some color class under all vertex colorings by colors. It has been previously shown that certain y!F) "large " order graphs G with "few " edge ..."
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A connected graph G is said to be Fgood if the Ramsey number r(F,G) has the value ( y(F)1) (p(G)1) + s(F), where s(F) is the minimum number of vertices in some color class under all vertex colorings by colors. It has been previously shown that certain y!F) "large " order graphs G with "few " edges are Fgood when F is a fixed multipartite graph. We show when F is a complete bipartite graph that this edge condition can be relaxed. 164 Burr, Erdtir, hudm, Rouueau, and Sch.lp Let F and G be (simple) graphs. The Ramsey number
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