Weight of edges in normal plane maps

The weight w ( e ) of an edge e in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge e = u v is an ( i , j ) -edge if d ( u ) ? i and d ( v ) ? j . In 1940, Lebesgue proved that every NPM has a ( 3 , 11 ) -edge, or ( 4 , 7 ) -edge, or ( 5 , 6 ) -edge, where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-polytope has an edge e with w ( e ) ? 13 , which bound is sharp. Borodin (1987), answering Erd?s' question, proved that every NPM has either a ( 3 , 10 ) -edge, or ( 4 , 7 ) -edge, or ( 5 , 6 ) -edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencova and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge e with w ( e ) ? 12 .The purpose of our note is to prove that every NPM has either a simplicial 3-vertex adjacent to a vertex degree at most 10, or ( 3 , 9 ) -edge, or ( 4 , 7 ) -edge, or ( 5 , 6 ) -edge. In particular, this confirms the above mentioned conjecture by Ferencova and Madaras. Furthermore, we construct a 3-polytope showing that the above term ( 3 , 9 ) is best possible.