  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 $(3,\leq23)$ or $(4,\leq11)$ or $(5,\leq9)$ or $(6,\leq8)$
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 Hud\'ak and \v{S}ugerek
[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 $(4,\leq 13)$ or $(5,\leq 9)$ or $(6,\leq 8)$ or $(7,
7)$.
