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arXiv · 2506.00878

The minimum size of maximal bipartite IC-plane graphs with given connectivity

Guiping Wang·Yuanqiu Huang·Zhangdong Ouyang·Licheng Zhang
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Abstract

Recently, the problem of establishing bounds on the edge density of 1-planar graphs, including their subclass IC-planar graphs, has received considerable attention. In 2018, Angelini et al. showed that any n-vertex bipartite IC-planar graph has at most 2.25n-4 edges, which implies that bipartite IC-planar graphs have vertex-connectivity at most 4. In this paper, we prove that any n-vertex maximal bipartite IC-plane graph with connectivity 2 has at least 3/2n-2 edges, and those with connectivity 3 has at least 2n-3 edges. All the above lower bounds are tight. For 4-connected maximal bipartite IC-planar graphs, the question of determining a non-trivial lower bound on the size remains open.

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Guiping Wang, Yuanqiu Huang, Zhangdong Ouyang, Licheng Zhang. 2025-06-01. The minimum size of maximal bipartite IC-plane graphs with given connectivity. https://arxiv.org/abs/2506.00878

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