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 for any k-crossing-critical graph G, cr(G)≤2.5k+16 holds, and in particular, if δ(G)≥4, then cr(G)≤2k+35 holds, where δ(G) is the minimum degree of G. In this paper, we improve these upper bounds to 2.5k+2.5 and 2k+8 respectively. In particular, for any k-crossing-critical graph G with n vertices, if δ(G)≥5, then cr(G)≤2k−k/2n+35/6 holds.
@article{arxiv.2003.06579,
title = {New upper bounds for the crossing numbers of crossing-critical graphs},
author = {Zongpeng Ding and Zhangdong Ouyang and Yuanqiu Huang and Fengming Dong},
journal= {arXiv preprint arXiv:2003.06579},
year = {2020}
}