2 research outputs found
A note on 1-planar graphs with minimum degree 7
It is well-known that 1-planar graphs have minimum degree at most 7, and not
hard to see that some 1-planar graphs have minimum degree exactly 7. In this
note we show that any such 1-planar graph has at least 24 vertices, and this is
tight.Comment: 4 page
Matchings in 1-planar graphs with large minimum degree
In 1979, Nishizeki and Baybars showed that every planar graph with minimum
degree 3 has a matching of size (where the constant 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 {\em 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 , 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