Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The absence of efficient dual pairs of spanning trees in planar graphs. | The absence of efficient dual pairs of spanning trees in planar graphs T. R. Riley and W. P. Thurston Mathematics Department 310 Malott Hall Cornell University Ithaca NY 14853-4201 USA wpt@ Submitted Dec 7 2005 Accepted Aug 18 2006 Published Aug 25 2006 2000 Mathematics Subject Classification 05C10 05C12 20F06 57M15 Abstract A spanning tree T in a finite planar connected graph G determines a dual spanning tree T in the dual graph G such that T and T do not intersect. We show that it is not always possible to find T in G such that the diameters of T and T are both within a uniform multiplicative constant independent of G of the diameters of their ambient graphs. 1 Introduction Suppose G is a finite connected undirected graph or multigraph embedded in the plane. Given a spanning tree T in G define T to be the spanning tree in the dual graph G whose edges are those dual to edges in G r T. Figure 1 gives an example. The length of a walk in a graph is the number of edges it contains and the distance between two vertices is the length of the shortest walk between them. The diameter The authors gratefully acknowledge support from NSF grants DMS-0540830 and DMS-0513436. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N13 1 Diam G of a finite connected graph G is the maximum distance between pairs of vertices of G. Motivated by issues arising in Geometric Group Theory concerning the geometry of van Kampen diagrams Gersten Riley asked 3 Question 1. Does there exists C 0 such that if G is a finite connected planar multi- graph then there is a maximal tree T in G with Diam T C Diam G and Diam T C Diam G They conjectured positive answers to a number of variants of this question with bounds imposed on the degrees of vertices in G or G . We exhibit a family of graphs resolving these negatively. Theorem 2. There are families Gn n N of finite connected planar graphs such that all vertices in Gn and Gn have degree at most 6 and there are .
