Large Matchings in Maximal 1-planar graphs

TL;DR

This paper establishes tight lower bounds on the size of maximum matchings in maximal 1-planar graphs, extending known results from maximal planar graphs to graphs with at most one crossing per edge.

Contribution

It provides new matching size bounds for maximal 1-planar graphs, including 3-connected and non-3-connected cases, with tight bounds demonstrated.

Findings

3-connected maximal 1-planar graphs have matchings of size at least (2n+6)/5

Non-3-connected maximal 1-planar graphs have matchings of size at least (3n+14)/10

Bounds are tight, with examples showing no larger matchings are possible under the restrictions.

Abstract

It is well-known that every maximal planar graph has a matching of size at least $\tfrac{n+8}{3}$ if $n\geq 14$. In this paper, we investigate similar matching-bounds for maximal \emph{1-planar} graphs, i.e., graphs that can be drawn such that every edge has at most one crossing. In particular we show that every 3-connected simple-maximal 1-planar graph has a matching of size at least $\tfrac{2n+6}{5}$; the bound decreases to $\tfrac{3n+14}{10}$ if the graph need not be 3-connected. We also give (weaker) bounds when the graph comes with a fixed 1-planar drawing or is not simple. All our bounds are tight in the sense that some graph that satisfies the restrictions has no bigger matching.