Journal of Combinatorial Theory, Series B 99 (2009) 97–109 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb On the number of 4-contractible edges in 4-connected graphs K. Ando a , Y. Egawa b , K. Kawarabayashi c , Matthias Kriesell d a Univ. Electro-Communications, 1-5-1 Chofugaoka Chofu-shi, Tokyo 182-8585, Japan b Science University of Tokyo, Shinjuku-ku, Tokyo 162, Japan c National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan d Mathematisches Seminar der Universität Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany article info abstract Article history: Received 4 June 2007 Available online 22 May 2008 Keywords: Connectivity Contractible edge Average degree We prove that every finite 4-connected graph G has at least 1 34 · (| E (G)|− 2| V (G)|) many contractible edges.  2008 Elsevier Inc. All rights reserved. 1. Introduction All graphs considered here are supposed to be finite, simple, and undirected. For terminology not defined here we refer to [1] or [2]. An edge e = xy in a k-connected graph G is called k-contractible if the graph G/e obtained from G identifying x, y and simplifying the result is k-connected. It is easy to see that every edge of a connected graph is 1-contractible, and it is a well-known fact that every vertex of a 2-connected graph nonisomorphic to K 3 is incident with a 2-contractible edge. The corresponding statement for 3-connected graphs fails, but it is still true that for an arbitrary vertex x in a 3-connected graph nonisomorphic to K 4 there is a 3-contractible edge at distance 0 or 1 from x (references in [5]). No such result holds for 4-connected graphs, as there are 4-connected graphs without 4- contractible edges; these are squares of cycles of length at least 5 and 4-connected line graphs of cubic graphs, and there are no other graphs without 4-contractible edges [3,9]. As they are all 4- regular, every 4-connected graph G whose average degree d(G) is larger than 4 must have at least one 4-contractible edge. E-mail address: kriesell@math.uni-hamburg.de (M. Kriesell). 0095-8956/$ – see front matter  2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jctb.2008.04.003