Optimal radio labellings of complete mm-ary trees

A radio labelling of a connected graph GG is a mapping f:V(G)→{0,1,2,…}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)u,v∈V(G), where diam(G) is the diameter of GG and d(u,v)d(u,v) the distance between uu and vv. The span of ff is defined as maxu,v∈V(G)|f(u)−f(v)|maxu,v∈V(G)|f(u)−f(v)|, and the radio number of GG is the minimum span of a radio labelling of GG. A complete mm-ary tree (m≥2m≥2) is a rooted tree such that each vertex of degree greater than one has exactly mm 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 mm-ary tree for any m≥2m≥2 with any height and construct explicitly an optimal radio labelling.