# Supersaturation Problem for the Bowtie

@article{Kang2017SupersaturationPF, title={Supersaturation Problem for the Bowtie}, author={Mihyun Kang and Tam{\'a}s Makai and Oleg Pikhurko}, journal={Electron. Notes Discret. Math.}, year={2017}, volume={61}, pages={679-685} }

The Tur\'an function $ex(n,F)$ denotes the maximal number of edges in an $F$-free graph on $n$ vertices. We consider the function $h_F(n,q)$, the minimal number of copies of $F$ in a graph on $n$ vertices with $ex(n,F)+q$ edges. The value of $h_F(n,q)$ has been extensively studied when $F$ is bipartite or colour-critical. In this paper we investigate the simplest remaining graph $F$, namely, two triangles sharing a vertex, and establish the asymptotic value of $h_F(n,q)$ for $q=o(n^2)$.

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#### References

SHOWING 1-10 OF 41 REFERENCES

Supersaturation problem for color-critical graphs

- Mathematics, Computer Science
- J. Comb. Theory, Ser. B
- 2017

This work proves that c1 > 0 for every color-critical F, and allows us to determine c1 for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge. Expand

Two Approaches to Sidorenko's Conjecture

- Mathematics
- 2013

Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots,k\}$, $\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|} \ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|}$ holds,… Expand

On the number of complete subgraphs of a graph II

- Mathematics
- 1983

Generalizing some results of P. Erdős and some of L. Moser and J. W. Moon we give lower bounds on the number of complete p-graphs K p of graphs in terms of the numbers of vertices and edges. Further,… Expand

Extremal Graph Problems , Degenerate Extremal Problems , and Supersaturated Graphs

- 2010

Notation. Given a graph, hypergraph Gn, . . . , the upper index always denotes the number of vertices, e(G), v(G) and χ(G) denote the number of edges, vertices and the chromatic number of G… Expand

Some theorems on graphs and posets

- Computer Science, Mathematics
- Discret. Math.
- 1976

It is proved that the poset P consisting of all induced connected subgraphs of a nontrivial connected graph G, partially ordered by inclusion, has dimension n where n is the number of noncut vertices in G whether or not P is a lattice. Expand

On the Minimal Density of Triangles in Graphs

- Mathematics, Computer Science
- Combinatorics, Probability and Computing
- 2008

This paper proves that g_3(\rho) is the minimal possible density of triangles in a graph with edge density ρ by proving that $t\df \lfloor 1/(1-\rho)\rfloor$ is the integer such that $\rho\in\bigl[ 1-\frac 1t,1- \frac 1{t+1}\bigr]$. Expand

SOME RECENT RESULTS ON EXTREMAL PROBLEMS IN GRAPH THEORY (Results)

- 2002

Three years ago I gave a talk on extremal problems in graph theory at Smolenice [2]. I will refer to this paper as I. I will only discuss results which have been found since the publication of I.… Expand

The clique density theorem

- Mathematics
- 2016

Tur an’s theorem is a cornerstone of extremal graph theory. It asserts that for any integer r > 2, every graph on n vertices with more than r 2 2(r 1) n 2 edges contains a clique of sizer, i.e.,r… Expand

On a problem of Turán

- Physics
- 1983

Let T(n, k +1, k) denote the least number of k-subsets of an n-set such that every k +1-subset contains on of the chosen k-sebsets. We show
$$ \mathop {\lim }\limits_{n \to \infty } \frac{{T(n,k +… Expand

Supersaturated graphs and hypergraphs

- Mathematics, Computer Science
- Comb.
- 1983

We shall consider graphs (hypergraphs) without loops and multiple edges. Let ℒ be a family of so called prohibited graphs and ex (n, ℒ) denote the maximum number of edges (hyperedges) a graph… Expand