COMBINATORICA Bolyai Society – Springer-Verlag 0209–9683/101/$6.00 c 2001 J´ anos Bolyai Mathematical Society Combinatorica 21 (3) (2001) 351–359 ON A MAX-MIN PROBLEM CONCERNING WEIGHTS OF EDGES STANISLAV JENDROL’, INGO SCHIERMEYER Received July 12, 1999 The weight w(e) of an edge e = uv of a graph is defined to be the sum of degrees of the vertices u and v. In 1990 P. Erd˝os asked the question: What is the minimum weight of an edge of a graph G having n vertices and m edges? This paper brings a precise answer to the above question of Erd˝os. 1. Introduction The weight w(e) of an edge e = uv of a graph G is defined to be the sum of degrees of the vertices u,v. This concept of the weight of an edge was introduced by Kotzig [8] who proved the following beautiful result: Every planar 3-connected graph contains an edge of weight not exceeding 13. This result was further developed in various directions. Gr¨ unbaum [4], Jucoviˇ c[7], Borodin [1], Fabrici and Jendrol’ [3] have studied inequalities for the number of edges having weight not exceeding 13 in planar 3-connected graphs. Ivanˇ co [5] has found an analogue of Kotzig’s result for graphs with minimum degree at least 3 and embedded on orientable 2-manifolds. The analogue of Kotzig’s result for triangulations of orientable 2-manifolds can be found in Zaks [9]. Recently Fabrici and Jendrol’ [3] proved that each 3-connected planar graph of maximum degree ≥ k contains a path on k vertices such that each of its vertices has degree at most 5k; the bound 5k being best possible. Enomoto and Ota [2] have proved that each planar 3-connected graph of Mathematics Subject Classification (2000): 05C35