On the radio number for corona of paths and cycles

Abstract

Radio k-coloring of graphs is one of the variations of frequency assignment problem. For a simple connected graph G and a positive integer k?diam(G), a radio k-coloring is an assignment f of positive integers (colors) to the vertices of G such that for every pair of distinct vertices u and v of G, the difference between their colors is at least 1+k?d(u,v). The maximum color assigned by f is called its span, denoted by rck(f). The radio k-chromatic number rck(G) of G is min{rck(f):fis a radiok-coloring ofG}. If d is the diameter of G, then a radio d-coloring is referred as a radio coloring and the radio d-chromatic number as the radio number, denoted by rn(G), of G. The corona G?H of two graphs G and H is the graph obtained by taking one copy of G and |V(G)| copies of H, and joining each and every vertex of the ith copy of H with the ith vertex of G by an edge. In this paper, for path Pn and cycle Cm, m?5, we determine rn(Pn?Cm) when n is even, and give an upper bound for the same when n is odd. Also, for m?4, we determine the radio number of Pn?Pm when n is even, and give both upper and lower bounds for rn(Pn?Pm) when n is odd. 2019 Kalasalingam University