Matchings in 1-planar graphs with large minimum degree

TL;DR

This paper establishes lower bounds on the size of matchings in 1-planar graphs with minimum degrees 3, 4, and 5, extending classical planar graph results to a broader class of graphs.

Contribution

It provides new matching size bounds for 1-planar graphs with minimum degrees 3, 4, and 5, including tight bounds for degree 3, and explores open cases for higher degrees.

Findings

Matching size at least n/7 + 12/7 for degree 3

Bounds established for degrees 4 and 5

Open problem for degrees 6 and 7

Abstract

In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size $\frac{n}{3}+c$ (where the constant $c$ depends on the connectivity), and even better bounds hold for planar graphs with minimum degree 4 and 5. In this paper, we investigate similar matching-bounds for {\em 1-planar} graphs, i.e., graphs that can be drawn such that every edge has at most one crossing. We show that every 1-planar graph with minimum degree 3 has a matching of size at least $\frac{1}{7}n+\frac{12}{7}$, and this is tight for some graphs. We provide similar bounds for 1-planar graphs with minimum degree 4 and 5, while the case of minimum degree 6 and 7 remains open.