Valid path-based graph vertex numbering

A labelling of a graph is an assignment of labels to its vertex or edge sets (or both), subject to certain conditions, a well established concept. A labelling of a graph G of order n is termed a numbering when the set of integers {1,...,n} is used to label the vertices of G distinctly. A 2-path (a path with three vertices) in a vertex-numbered graph is said to be valid if the number of its middle vertex is smaller than the numbers of its endpoints. The problem of finding a vertex numbering of a given graph that optimises the number of induced valid 2-paths is studied, which is conjectured to be in the NP-hard class. The reported results for several graph classes show that apparently there are not one or more numbering patterns applicable to different classes of graphs, which requires the development of a specific numbering for each graph class under study

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 conjectured the solution to the problem for 5-cycles. We confirm their conjecture

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

Let $f(n,H)$ denote the maximum number of copies of $H$ possible in an $n$-vertex planar graph. The function $f(n,H)$ has been determined when $H$ is a cycle of length $3$ or $4$ by Hakimi and Schmeichel and when $H$ is a complete bipartite graph with smaller part of size 1 or 2 by Alon and Caro. We determine $f(n,H)$ exactly in the case when $H$ is a path of length 3.Comment: A simpler proof of the main result is now give

Subgraph densities in a surface

Given a fixed graph $H$ that embeds in a surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\Sigma$? We show that the answer is $\Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.Comment: v4: referee's comments implemented. v3: proof of the main theorem fully rewritten, fixes a serious error in the previous version found by Kevin Hendre

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, and we identify the graphs which achieve the maximum in the C5 case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically