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Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n/{log}_2 n)$ separating 4-cycles then $G$ has $Omega(n^2)$ Hamiltonian cycles, and if $delta(G)ge 5$ then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a double wheel structure, providing further evidence to the above conjecture.
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Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles.
For a fixed planar graph $H$, let $operatorname{mathbf{N}}_{mathcal{P}}(n,H)$ 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 $operatorname{mathbf{N}}_{mathcal{P}}(n,C_m)$ is currently known for $min{3,4,5,6,8}$. In this note, we extend this list by establishing $operatorname{mathbf{N}}_{mathcal{P}}(n,C_{10})sim(n/5)^5$ and $operatorname{mathbf{N}}_{mathcal{P}}(n,C_{12})sim(n/6)^6$. We prove this by answering the following question for $min{5,6}$, which is interesting in its own right: which probability mass $mu$ on the edges of some clique maximizes the probability that $m$ independent samples from $mu$ form an $m$-cycle?
Hakimi and Schmeichel determined a sharp lower bound for the number of cycles of length 4 in a maximal planar graph with $n$ vertices, $ngeq 5$. It has been shown that the bound is sharp for $n = 5,12$ and $ngeq 14$ vertices. However, it was only conjectured by the authors about the minimum number of cycles of length 4 for maximal planar graphs with the remaining small vertex numbers. In this note we confirm their conjecture.
We show that the number of partial triangulations of a set of $n$ points on the plane is at least the $(n-2)$-nd Catalan number. This is tight for convex $n$-gons. We also describe all the equality cases.
We prove that if $G$ is a $k$-partite graph on $n$ vertices in which all of the parts have order at most $n/r$ and every vertex is adjacent to at least a $1-1/r+o(1)$ proportion of the vertices in every other part, then $G$ contains the $(r-1)$-st power of a Hamiltonian cycle
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