Light edges in 1-planar graphs of minimum degree 3
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A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least 3 contains an edge with degrees of its endvertices of type (3,≤23) or (4,≤11) or (5,≤9) or (6,≤8) or (7,7). Moreover, the upper bounds 9,8 and 7 here are sharp and the upper bounds 23 and 11 are very close to the possible sharp ones, which may be 20 and 10, respectively. This generalizes a result of Fabrici and Madaras [Discrete Math., 307 (2007) 854--865] which says that each 3-connected 1-planar graph contains a light edge, and improves a result of Hudák and Šugerek [Discuss. Math. Graph Theory, 32(3) (2012) 545--556], which states that each 1-planar graph of minimum degree at least 4 contains an edge with degrees of its endvertices of type (4,≤ 13) or (5,≤ 9) or (6,≤ 8) or (7, 7).
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