A Theorem About a Contractible and Light Edge

In 1955 Kotzig proved that every planar 3-connected graph contains an edge such that sum of degrees of its endvertices is at most 13. Moreover, if the graph does not contain 3-vertices, then this sum is at most 11. Such an edge is called light. The well-known result of Steinitz that the 3-connected planar graphs are precisely the skeletons of 3-polytopes, gives an additional trump to Kotzig’s theorem. On the other hand, in 1961, Tutte proved that every 3-connected graph, distinct from K4, contains a contractible edge. In this paper, we strengthen Kotzig’s theorem by showing that every 3-connected planar graph distinct from K4 contains an edge which is both light and contractible. A consequence is that every 3-polytope can be constructed from the Tetrahedron by a sequence of splittings of vertices of degree at most 11. ∗Supported in part by project LN00A056 of the Czech Ministry of Education. †Supported in part by Ministry of Science and Technology of Slovenia, Research Project Z13129.