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On local structure of 1-planar graphs of minimum degree 5 and girth 4. (English) Zbl 1194.05025
Summary: A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains
(1) a 5-vertex adjacent to an \(\leq6\)-vertex,
(2) a 4-cycle whose every vertex has degree at most 9,
(3) a \(K_{1,4}\) with all vertices having degree at most 11.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
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