tailieunhanh - Báo cáo toán học: "Edge-magic group labellings of countable graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Edge-magic group labellings of countable graphs. | Edge-magic group labellings of countable graphs Nicholas Cavenagh and Diana Combe School of Mathematics and Statistics The University of New South Wales Sydney NSW 2052 Australia nickc@ diana@ Adrian M. Nelson School of Mathematics and Statistics University of Sydney NSW 2006 Australia adriann@ Submitted May 18 2006 Accepted Sep 22 2006 Published Oct 27 2006 Mathematics Subject Classification 05C78 Abstract We investigate the existence of edge-magic labellings of countably infinite graphs by abelian groups. We show for that for a large class of abelian groups including the integers Z there is such a labelling whenever the graph has an infinite set of disjoint edges. A graph without an infinite set of disjoint edges must be some subgraph of H I where H is some finite graph and I is a countable set of isolated vertices. Using power series of rational functions we show that any edge-magic Z-labelling of H 1 has almost all vertex labels making up pairs of half-modulus classes. We also classify all possible edge-magic Z-labellings of H 1 under the assumption that the vertices of the finite graph are labelled consecutively. 1 Introduction. By countable we mean countably infinite. Our graphs have no loops and no multiple edges. The vertex set is non-empty and is denoted V. The edge set E is a possibly empty set of unordered pairs of vertices. An edge fx yg is usually denoted xy or yx . The set V u E is the set of graph elements. When we say a graph is countable we mean that the set of graph elements is countable and hence that the vertex set is countable and the edge set is finite or countable. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R92 1 In this paper the group A is always a countable abelian group. Since we are often considering the integers Z it is convenient to consider our groups additively. For a countable graph G an A-labelling of G or a labelling of G over A is a bijection from V u E to A. For any group .