Large matchings in maximal 1-planar graphs

It is well-known that every maximal planar graph has a matching of size at least n + 8 3 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 2 n + 6 5 ; the bound decreases to 3 n + 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.