Optimal radio labellings of complete m-ary trees

  • Authors:

  • Xiangwen Li;Vicky Mak;Sanming Zhou

  • Affiliations:

  • Department of Mathematics, Huazhong Normal University, Wuhan 430079, China and Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia;School of Information Technology, Deakin University, VIC 3125, Australia;Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

A radio labelling of a connected graph G is a mapping f:V(G)-{0,1,2,...} such that |f(u)-f(v)|=diam(G)-d(u,v)+1 for each pair of distinct vertices u,v@?V(G), where diam(G) is the diameter of G and d(u,v) the distance between u and v. The span of f is defined as max"u","v"@?"V"("G")|f(u)-f(v)|, and the radio number of G is the minimum span of a radio labelling of G. A complete m-ary tree (m=2) is a rooted tree such that each vertex of degree greater than one has exactly m children and all degree-one vertices are of equal distance (height) to the root. In this paper we determine the radio number of the complete m-ary tree for any m=2 with any height and construct explicitly an optimal radio labelling.