Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later, Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles. Along this direction, Brinkmann, Souffriau, and Van Cleemput in 2018 established a linear lower bound on the number of hamiltonian cycles in 4-connected planar triangulations. In stark contrast, Alahmadi, Aldred, and Thomassen in 2020 showed that every 5-connected triangulation of the plane or the projective plane has exponentially many hamiltonian cycles. This gives the motivation to study the number of hamiltonian cycles of 4-connected triangulations with few 4-separators. Recently, Liu and Yu in 2021 showed that every 4-connected planar triangulation with $O(n / \log n)$ 4-separators has a quadratic number of hamiltonian cycles. By adapting the framework of Alahmadi, Aldred, and Thomassen, we strengthen the last two aforementioned results. We prove that every 4-connected planar or projective planar triangulation with $O(n)$ 4-separators has exponentially many hamiltonian cycles.

For any positive integer $k$, define $f(k)$ (respectively, $f_3(k)$) to be the minimal integer $\ge k$ such that every 3-connected planar graph $G$ (respectively, 3-connected cubic planar graph $G$) of circumference $\ge k$ has a cycle whose length is in the interval $[k, f(k)]$ (respectively, $[k, f_3(k)]$). Merker showed that $f_3(k) \le 2k + 9$ for any $k \ge 2$, and $f_3(k) \ge 2k + 2$ for any even $k \ge 4$. He conjectured that $f_3(k) \le 2k + 2$ for any $k \ge 2$. This conjecture was disproved by Zamfirescu, who gave an infinite family of counterexamples for every even $k \ge 6$ whose graphs have no cycle length in $[k, 2k + 2]$, i.e., $f_3(k) \ge 2k + 3$ for any even $k \ge 6$. However, the exact value of $f_3(k)$ was only known for $k \le 4$, and it is a natural problem to determine $f_3(k)$ for $k \ge 5$. In this paper, we improve Merker's upper bound, and give the exact value of $f_3(k)$ for every $k \ge 5$. We show that $f_3(5) = 10$, $f_3(7) = 15$, $f_3(9) = 20$, and $f_3(k) = 2k + 3$ for any $k = 6, 8$ or $\ge 10$. For general 3-connected planar graphs, Merker conjectured that there exists some positive integer $c$ such that $f(k) \le 2k + c$ for any positive integer $k$. We give a complete positive answer to this conjecture. We prove that $f(k) = 5$ for any $k \le 3$, $f(4) = 10$, and $f(k) = 2k + 3$ for any $k \ge 5$.

Two vertices v and w of a graph G are called a pendant pair if the maximal number of edge-disjoint paths in G between them is precisely min{d(v),d(w)}, where d denotes the degree function. The importance of pendant pairs stems from the fact that they are the key ingredient in one of the simplest and most widely used algorithms for the minimum cut problem today. Mader showed 1974 that every simple graph with minimum degree delta contains Omega(delta^2) pendant pairs; this is the best bound known so far. We improve this result by showing that every simple graph G with minimum degree delta >= 5 or with edge-connectivity lambda >= 4 or with vertex-connectivity kappa >= 3 contains in fact Omega(delta |V|) pendant pairs. We prove that this bound is tight from several perspectives, and that Omega(delta |V|) pendant pairs can be computed efficiently, namely in linear time when a Gomory-Hu tree is given. Our method utilizes a new cut tree representation of graphs.