Tight gaps in the cycle spectrum of 3-connected planar graphs

TL;DR

This paper determines exact cycle length bounds in 3-connected planar graphs and cubic variants, confirming conjectures and disproving others, thus advancing understanding of cycle spectra in these graphs.

Contribution

The paper establishes exact values for the cycle spectrum bounds in 3-connected planar and cubic planar graphs, resolving open questions and confirming conjectures in the field.

Findings

Exact values of f_3(k) for k ≥ 5 are determined.

Confirmed that f(k) = 2k + 3 for k ≥ 5.

Proved that f(k) = 5 for k ≤ 3 and f(4) = 10.

Abstract

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 was left open 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 $k \ge 5$.