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On local properties of 1-planar graphs with high minimum degree. (English) Zbl 1237.05053
Summary: A graph is called 1-planar if there exists its drawing in the plane such that each edge contains at most one crossing. We prove that each 1-planar graph of minimum degree 7 contains a pair of adjacent vertices of degree 7 as well as several small graphs whose vertices have small degrees; we also prove the existence of a 4-cycle with relatively small degree vertices in 1-planar graphs of minimum degree at least 6.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C62 Graph representations (geometric and intersection representations, etc.)
05C07 Vertex degrees
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