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} }
9 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…
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.
Generalized Planar Turán Numbers
- MathematicsElectron. J. Comb.
- 2021
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…
References
SHOWING 1-10 OF 26 REFERENCES
On the number of cycles of length k in a maximal planar graph
- MathematicsJ. Graph Theory
- 1979
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 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…
On The Number of Subgraphs of Prescribed Type of Planar Graphs With A Given Number of Vertices
- Mathematics
- 1984
The Maximum Number of Pentagons in a Planar Graph
- Mathematics
- 2019
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…
A note on the maximum number of triangles in a C5‐free graph
- MathematicsJ. Graph Theory
- 2019
We prove that the maximum number of triangles in a C 5 ‐free graph on n vertices is at most 122(1+o(1))n3∕2 , improving an estimate of Alon and Shikhelman.
The Maximum Number of Triangles in C2k+1-Free Graphs
- MathematicsCombinatorics, Probability and Computing
- 2012
Upper and lower bounds are proved for the maximum number of triangles in C2k+1-free graphs. The bounds involve extremal numbers related to appropriate even cycles.
Triangles in
C
5 ‐free graphs and hypergraphs of girth six
- MathematicsJ. Graph Theory
- 2022
It is proved that the maximum number of triangles in a C_5-free graph on n vertices is at most 1 + o(1) n 3/2, and a connection to r-uniform hypergraphs without (Berge) cycles of length less than six is shown.
On the maximum number of five-cycles in a triangle-free graph
- MathematicsJ. Comb. Theory, Ser. B
- 2012