Optimal radio labelings of the Cartesian product of the generalized Peterson graph and tree

Abstract

A radio labeling of a graph [Formula: see text] is a function [Formula: see text] such that [Formula: see text] for every pair of distinct vertices [Formula: see text] of [Formula: see text]. The radio number of [Formula: see text], denoted by [Formula: see text], is the smallest number [Formula: see text] such that [Formula: see text] has radio labeling [Formula: see text] with max[Formula: see text] = [Formula: see text]. In this paper, we give a lower bound for the radio number for the Cartesian product of the generalized Petersen graph and tree. We present two necessary and sufficient conditions, and three other sufficient conditions to achieve the lower bound. Using these results, we determine the radio number for the Cartesian product of the Peterson graph and stars.