Optimal radio labellings of complete m-ary trees

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: Keywords: Channel assignment Radio labelling Radio number m-ary tree Binary tree a b s t r a c t 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.