Improved lower bounds for the radio number of trees

Abstract

Let G be a graph with diameter d. A radio labelling of G is a function f that assigns to each vertex with a non-negative integer such that the following holds for all vertices u,v: |f(u)−f(v)|⩾d+1−d(u,v), where d(u,v) is the distance between u and v. The span of f is the absolute difference of the largest and smallest values in f(V). The radio number of G is the minimum span of a radio labelling admitted by G. For trees, a general lower bound of the radio number was given by Liu [10], which has been used to prove special families of trees whose radio number is equal to this bound [1,6,9,10]. In this article, we present improved lower bounds for some trees whose radio number exceeds Liu's lower bound. Some of these new bounds are sharp for special families of trees, including complete binary trees [9] and odd paths [11]. Moreover, using these new bounds, we extend the known results of Halasza and Tuza [6] on complete level-wise regular trees.