Linkage problem on optimal $1$-planar graphs

Shohei Koizumi; Ryosuke Osaka; Yusuke Suzuki

Linkage problem on optimal $1$-planar graphs

Combinatorics 2026-06-26 v1

Authors: Shohei Koizumi , Ryosuke Osaka , Yusuke Suzuki

Abstract

Enami and Maezawa give a complete characterization of $(s_1, s_2, \ldots, s_k)$ -linked planar graphs for any $k$ -tuple of positive integers. In this paper, we investigate linkage problems for optimal 1-planar graphs. In particular, we show that every optimal 1-planar graph with connectivity $6$ is $(5, 5)$ -linked. Moreover, for an optimal $1$ -planar graph $G$ that is not $(2,2,1)$ -linked, we characterize disjoint vertex subsets $S_1, S_2, S_3$ in $G$ with $|S_1|=|S_2|=2$ and $|S_3|=1$ such that $G$ is not $\{S_1,S_2,S_3\}$ -linked.

Cite

@article{arxiv.2606.27682,
  title  = {Linkage problem on optimal $1$-planar graphs},
  author = {Shohei Koizumi and Ryosuke Osaka and Yusuke Suzuki},
  journal= {arXiv preprint arXiv:2606.27682},
  year   = {2026}
}