On the size of matchings in 1-planar graph with high minimum degree

TL;DR

This paper establishes a tight lower bound on the size of matchings in 1-planar graphs with minimum degree 6, confirming a conjecture and extending known results for graphs with lower minimum degrees.

Contribution

It proves a new tight lower bound for matchings in 1-planar graphs with minimum degree 6, advancing understanding of their structural properties.

Findings

Proves that such graphs have matchings of size at least (3n+4)/7.

Confirms the conjecture by Biedl and Wittnebel for minimum degree 6.

Extends known bounds for graphs with lower minimum degrees.

Abstract

A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph with minimum degree 3 and $n\geq 7$ vertices has a matching of size at least $\frac{n+12}{7}$, which is tight for some graphs. They also provided tight lower bounds for the sizes of matchings in 1-planar graphs with minimum degree 4 or 5. In this paper, we show that any 1-planar graph with minimum degree 6 and $n \geq 36$ vertices has a matching of size at least $\frac{3n+4}{7}$, and this lower bound is tight. Our result confirms a conjecture posed by Biedl and Wittnebel.