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18 February 2025
 
  » arxiv » 2301.01394

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Large Matchings in Maximal 1-planar graphs
Therese Biedl ; John Wittnebel ;
Date 4 Jan 2023
AbstractIt is well-known that every maximal planar graph has a matching of size at least $ frac{n+8}{3}$ if $ngeq 14$. 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 $ frac{2n+6}{5}$; the bound decreases to $ frac{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.
Source arXiv, 2301.01394
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