Summary: T. Nishizeki and I. Baybars [Discrete Math. 28, 255–267 (1979; Zbl 0425.05032)] showed that every planar graph with minimum degree 3 has a matching of size \(\frac{n}{3} + c\) (where the constant \(c\) depends on the connectivity), and even better bounds hold for planar graphs with minimum degree 4 and 5. In this paper, we investigate similar matching-bounds for 1-planar graphs, that is, graphs that can be drawn such that every edge has at most one crossing. We show that every 1-planar graph with minimum degree 3 has a matching of size at least \(\frac{1}{7} n + \frac{12}{7}\), and this is tight for some graphs. We provide similar bounds for 1-planar graphs with minimum degree 4 and 5, while the case of minimum degree 6 and 7 remains open.
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