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

TL;DR

This paper investigates bounds on the number of edges in maximal bipartite IC-plane graphs with specific connectivity levels, providing tight bounds for connectivity 2 and 3, and highlighting open problems for connectivity 4.

Contribution

It establishes tight lower bounds on the size of maximal bipartite IC-plane graphs with connectivity 2 and 3, advancing understanding of their structural properties.

Findings

Connectivity 2 graphs have at least 1.5n-2 edges.

Connectivity 3 graphs have at least 2n-3 edges.

Bounds are tight for connectivity 2 and 3.

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.