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arXiv · 2207.03747

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

Yuanqiu Huang·Zhangdong Ouyang·Fengming Dong
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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.

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Yuanqiu Huang, Zhangdong Ouyang, Fengming Dong. 2022-07-08. On the size of matchings in 1-planar graph with high minimum degree. https://arxiv.org/abs/2207.03747

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