Near-perfect matchings in highly connected 1-planar graphs with a local crossing constraint

TL;DR

This paper proves that 6-connected 1-planar graphs with a specific local crossing structure always have near-perfect matchings, extending known results from planar graphs to a broader class.

Contribution

It establishes that 6-connected 1-planar graphs with a local crossing constraint have near-perfect matchings and a scattering number at most one, showing the optimality of 6-connectivity.

Findings

6-connected 1-planar graphs with local crossing constraints have near-perfect matchings

Their scattering number is at most one

6-connectivity is shown to be the best possible under these conditions

Abstract

For planar graphs, it is well known that high connectivity implies a Hamiltonian cycle and hence any 4-connected planar graph has a near-perfect matching. Nevertheless, whether 6-connected 1-planar graphs admit near-perfect matchings remains largely open. The prior art established this for 4-connected 1-planar graphs only when each crossing involves four endpoints that induce a $K_4$. In this paper, we study 6-connected 1-planar graphs that are drawn such that at all crossings the four endpoints induce a 4-cycle (plus perhaps more edges). We show that these have a near-perfect matching, and in fact even stronger, their scattering number is at most one. Moreover, under the local crossing restriction, the requirement of 6-connectivity is best possible; this is witnessed by explicit constructions due to Biedl and Fabrici et al.