# The maximum number of paths of length four in a planar graph

@article{Ghosh2021TheMN, title={The maximum number of paths of length four in a planar graph}, author={Debarun Ghosh and Ervin Gy{\"o}ri and Ryan R. Martin and Addisu Paulos and Nika Salia and Chuanqi Xiao and Oscar Zamora}, journal={Discret. Math.}, year={2021}, volume={344}, pages={112317} }

## 11 Citations

The Maximum Number of Paths of Length Three in a Planar Graph

- Mathematics
- 2019

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…

Planar graphs with the maximum number of induced 4-cycles or 5-cycles

- Mathematics
- 2021

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,…

Generalized outerplanar Tur\'an number of short paths

- Mathematics
- 2021

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…

The maximum number of 10- and 12-cycles in a planar graph

- Mathematics
- 2021

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

- Mathematics
- 2021

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…

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

- Mathematics
- 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…

Bounding the number of odd paths in planar graphs via convex optimization

- Mathematics, Computer Science
- 2022

This paper improves upon the previously best known bound by a factor e m, which is best possible up to the hidden constant, and makes a signiﬁcant step towards resolving conjectures of Gosh et al. and of Cox and Martin.

On the maximum number of edges in planar graphs of bounded degree and matching number

- Mathematics
- 2022

We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.

The maximum number of paths of length three in a planar graph

- MathematicsJournal of Graph Theory
- 2022

Subgraph densities in a surface

- MathematicsCombinatorics, Probability and Computing
- 2022

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.

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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…

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