Tight Gaps in the Cycle Spectrum of 3-Connected Planar Graphs

Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap

“…Remark. After this note was submitted, the article [1] by Cui and Lo appeared. It extends the results given in this note and fully settles the problem by determining tight gaps for cycle spectra in the class of 3-connected cubic planar graphs.…”

Tight cycle spectrum gaps of cubic 3-connected toroidal graphs

“…where one derives the first inequality as in [2]. This implies that n ≤ (6k − 4)x 2k − 1 < 3x, which contradicts (1).…”

“…However, in a short note the second author proved that σ 3 0 (k) ≥ 2k + 3 for all even k ≥ 6, see [5]. Cui and the first author [2] expanded on this and gave a full description of the values of σ 0 (k) and σ 3 0 (k) for all k; in particular, they showed that σ 0 (k) = σ 3 0 (k) = 2k + 3 for all k ≥ 10. The purpose of this note is to treat analogues of these results pertaining to the cubic toroidal case.…”