Counting paths, cycles, and blow‐ups in planar graphs

@inproceedings{Cox2022CountingPC,
  title={Counting paths, cycles, and blow‐ups in planar graphs},
  author={Christopher Cox and Ryan R. Martin},
  year={2022}
}
For a planar graph H , let N P ( n, H ) denote the maximum number of copies of H in an n -vertex planar graph. In this paper, we prove that N P ( n, P 7 ) ∼ 427 n 4 , N P ( n, C 6 ) ∼ ( n/ 3) 3 , N P ( n, C 8 ) ∼ ( n/ 4) 4 and N P ( n, K 4 { 1 } ) ∼ ( n/ 6) 6 , where K 4 { 1 } is the 1-subdivision of K 4 . In addition, we obtain significantly improved upper bounds on N P ( n, P 2 m +1 ) and N P ( n, C 2 m ) for m ≥ 4. For a wide class of graphs H , the key technique developed in this paper… 
The maximum number of 10- and 12-cycles in a planar graph
For a fixed planar graph H , let NP(n,H) denote the maximum number of copies of H in an n-vertex planar graph. In the case when H is a cycle, the asymptotic value of NP(n,Cm) is currently known for m
Planar graphs with the maximum number of induced 6-cycles
For large n we determine the maximum number of induced 6-cycles which can be contained in a planar graph on n vertices, and we classify the graphs which achieve this maximum. In particular we show
Generalized outerplanar Tur\'an number of short paths
Let H be a graph. The generalized outerplanar Turán number of H, denoted by fOP(n,H), is the maximum number of copies of H in an n-vertex outerplanar graph. Let Pk be the path on k vertices. In this
Generalized Planar Turán Numbers
In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán
The Maximum Number of Paths of Length Three in a Planar Graph
Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path of length $k$, is
The maximum number of induced C5 's in a planar graph
Finding the maximum number of induced cycles of length k in a graph on n vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidický and Pfender
Planar graphs with the maximum number of induced 4-cycles or 5-cycles
For large n we determine exactly the maximum numbers of induced C4 and C5 subgraphs that a planar graph on n vertices can contain. We show that K2,n−2 uniquely achieves this maximum in the C4 case,

References

SHOWING 1-10 OF 10 REFERENCES
On the number of cycles of length k in a maximal planar graph
TLDR
Bounds are given for Ck(G) when 5 ≤ k ≤ p, and in particular bounds for Cp(G), in terms of p are considered, in particular for C3 and C4.
The Maximum Number of Pentagons in a Planar Graph
Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an $n$-vertex planar graph. They determined this number exactly for triangles and 4-cycles and
Matrix analysis
TLDR
This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.
Foundations of Optimization
TLDR
This chapter discusses three Basic Optimization Algorithms, Duality Theory and Convex Programming, Semi-infinite Programming, and Topics in ConveXity.
Subgraph densities in a surface
TLDR
The answer to the maximum number of copies of H in an n-vertex graph is shown, which simultaneously answers two open problems posed by Eppstein.
The maximum number of paths of length four in a planar graph
The Maximum Number of Paths of Length Three in a Planar Graph
Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path of length $k$, is
Introduction to graph theory. Pearson, United States
  • 2018
Introduction to graph theory