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Counting cycles in planar triangulations
Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2024-11-05 , DOI: 10.1016/j.jctb.2024.10.002
On-Hei Solomon Lo, Carol T. Zamfirescu

We investigate the minimum number of cycles of specified lengths in planar n-vertex triangulations G. We prove that this number is Ω(n) for any cycle length at most 3+max{rad(G),(n32)log32}, where rad(G) denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian n-vertex triangulations containing O(n) many k-cycles for any k{nn5,,n}. Furthermore, we prove that planar 4-connected n-vertex triangulations contain Ω(n) many k-cycles for every k{3,,n}, and that, under certain additional conditions, they contain Ω(n2)k-cycles for many values of k, including n.

中文翻译:


平面三角测量中的周期计数



我们研究了平面 n 顶点三角剖分 G 中指定长度的最小循环数。我们证明,对于任何周期长度,这个数字都是 Ω(n),最大 3+max{rad(G⁎),⌈(n−32)log32⌉},其中 rad(G⁎) 表示三角剖分对偶的半径,它至少是对数的,但可以按照三角剖分的顺序是线性的。我们还表明,对于任何 k∈{⌈n−n5⌉,...,n},存在包含 O(n) 多个 k 周期的平面哈密顿 n 顶点三角剖分。此外,我们证明平面 4 连通 n 顶点三角剖分对于每个 k∈{3,...,n} 包含 Ω(n) 个周期,并且在某些附加条件下,它们包含 Ω(n2)k 个周期,用于 k 的许多值,包括 n。
更新日期:2024-11-05
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