Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Geodetic topological cycles in locally finite graphs. | Geodetic topological cycles in locally finite graphs Agelos Georgakopoulos Mathematisches Seminar Universităt Hamburg Bundesstrafie 55 20146 Hamburg Germany georgakopoulos@ Philipp Sprussel Mathematisches Seminar Universităat Hamburg Bundesstrafie 55 20146 Hamburg Germany spruessel@ Submitted Oct 31 2008 Accepted Nov 23 2009 Published Nov 30 2009 Mathematics Subject Classification 05C63 Abstract We prove that the topological cycle space C G of a locally finite graph G is generated by its geodetic topological circles. We further show that although the finite cycles of G generate C G its finite geodetic cycles need not generate C G . 1 Introduction A finite cycle C in a graph G is called geodetic if for any two vertices x y G C the length of at least one of the two x-y arcs on C equals the distance between x and y in G. It is easy to prove see Section Proposition . The cycle space of a finite graph is generated by its geodetic cycles. Our aim is to generalise Proposition to the topological cycle space of locally finite infinite graphs. The topological cycle space C G of a locally finite graph G was introduced by Diestel and Ktihn 10 11 . It is built not just from finite cycles but also from infinite circles homeomorphic images of the unit circle S1 in the topological space G consisting of G seen as a 1-complex together with its ends. See Section 2 for precise definitions. This space C G has been shown 2 3 4 5 10 15 to be the appropriate notion of the cycle space for a locally finite graph it allows generalisations to locally finite graphs of most of the well-known theorems about the cycle space of finite graphs theorems which fail for Supported by GIF grant no. . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R144 1 infinite graphs if the usual finitary notion of the cycle space is applied. It thus seems that the topological cycle space is an important object that merits further investigation. See 6 7 for .