Geometriae Dedicata 68: 91–99, 1997. 91 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. A Structural Property of Convex 3-Polytopes STANISLAV JENDROL’ Department of Geometry and Algebra, P. J. ˇ Saf´ arik University, Jesenn´ a 5, 041 54 Koˇ sice, Slovak Republic; e-mail: jendrol@kosice.upjs.sk (Received: 28 October 1996) Abstract. In this paper we prove that each convex 3-polytope contains a path on three vertices with restricted degrees which is one of the ten types. This result strengthens a theorem by Kotzig that each convex 3-polytope has an edge with the degree sum of its end vertices at most 13. Mathematics Subject Classifications (1991): 52B10, 52B05, 05C10, 05C38. Key words: convex 3-polytope, path, 3-connected planar graph. 1. Introduction This paper deals with graphs of convex 3-polytopes, this is, 3-connected planar graphs (see Steinitz’s theorem in [2]). We use standard terminology and notations according to Gr¨ unbaum’s book [2]. We recall, however, more specialized notations. A vertex of degree is called the -vertex, an edge joining an -vertex with a - vertex is called the -edge. A path on vertices 1 2 is defined to be the 1 2 -path provided that 12 ; denotes the degree of the vertex . It is an old classical consequence of the famous Euler’s formula that every convex 3-polytope contains a vertex of degree at most 5. A beautiful result of Kotzig [7], [8] (see also [3], [4], [5], [9]) states that each convex 3-polytope contains an -edge with 13, the bound being the best possible. Franklin [1] proved that each simplicial 3-polytope (all faces are triangles) of minimum degree 5 has an 5 -path, where 5 6. The present author [6] has proved that each convex 3-polytope has an -path and an -path for which max 15 (the bound is sharp) and max 23, respectively. The aim of this paper is to prove the following strengthening of the above- mentioned results by Kotzig, Franklin and the author. THEOREM 1. Every convex 3-polytope contains an -path where i 3 10 3 3 10 or ii 4 7 4 4 7 or