Large Matchings in Maximal 1-planar graphs

01/04/2023

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by   Therese Biedl, et al.

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It is well-known that every maximal planar graph has a matching of size at least n+83 if n≥ 14. In this paper, we investigate similar matching-bounds for maximal 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 2n+65; the bound decreases to 3n+1410 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.