Matchings in 1-planar graphs with large minimum degree
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In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size 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, i.e., 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 1/7n+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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