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New upper bounds for the crossing numbers of crossing-critical graphs
Citation
Ding, Z., Ouyang, Z., Huang, Y., & Dong, F. M. (2022). New upper bounds for the crossing numbers of crossing-critical graphs. Discrete Mathematics, 345(12), Article 113090. https://doi.org/10.1016/j.disc.2022.113090
Abstract
A graph G is k-crossing-critical if cr(G) ≥ k, but cr(G \ e) < k for each edge e ∈ E(G), where cr(G) is the crossing number of G. It is known that the latest upper bound of cr(G) for a k-crossing-critical graph G is 2k+8 √ k+47 when δ(G) ≥ 3, and 2k+35 when δ(G) ≥ 4, where δ(G) is the minimum degree of G. In this paper, we mainly show that for any k-crossing-critical graph G with n vertices, cr(G) ≤ 2k+8 when δ(G) ≥ 4, and cr(G) ≤ 2k− √ k/2n+35/6 when δ(G) ≥ 5.
Date Issued
2022
Publisher
Elsevier
Journal
Discrete Mathematics