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arXiv · 2301.01394

Large Matchings in Maximal 1-planar graphs

Therese Biedl·John Wittnebel
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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.

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Therese Biedl, John Wittnebel. 2023-01-04. Large Matchings in Maximal 1-planar graphs. https://arxiv.org/abs/2301.01394

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