New results on radio k-labelings of distance graphs

Abstract

A radio k -labeling of a graph G is an assignment of non-negative integers (labels) to the vertices of G such that the absolute value of the difference between the labels of any two distinct vertices u and v is not less than k + 1 − d ( u , v ) , where d ( u , v ) denotes the distance between u and v . The radio k -labeling number of G is the minimum span of a radio k -labeling of G . For a given set of integers D = { d 1 , d 2 , … , d t } , the distance graph G ( Z , D ) , denoted also by D ( d 1 , d 2 , … , d t ) , has the set of integers Z as its vertex set, while two distinct vertices i , j ∈ Z are adjacent in G ( Z , D ) if and only if | i − j | ∈ D . Radio k -labelings of three families of distance graphs are considered. Some improved theoretical lower and upper bounds on the radio k -labeling number are presented and some computer-based methods for computing radio k -labelings of these families of graphs are proposed.