On the radio number for corona of paths and cycles

Abstract

Radio (Formula presented.) -coloring of graphs is one of the variations of frequency assignment problem. For a simple connected graph (Formula presented.) and a positive integer (Formula presented.), a radio (Formula presented.) -coloring is an assignment (Formula presented.) of positive integers (colors) to the vertices of (Formula presented.) such that for every pair of distinct vertices (Formula presented.) and (Formula presented.) of (Formula presented.), the difference between their colors is at least (Formula presented.). The maximum color assigned by (Formula presented.) is called its span, denoted by (Formula presented.). The radio (Formula presented.) -chromatic number (Formula presented.) of (Formula presented.) is (Formula presented.). If (Formula presented.) is the diameter of (Formula presented.), then a radio (Formula presented.) -coloring is referred as a radio coloring and the radio (Formula presented.) -chromatic number as the radio number, denoted by (Formula presented.), of (Formula presented.). The corona (Formula presented.) of two graphs (Formula presented.) and (Formula presented.) is the graph obtained by taking one copy of (Formula presented.) and (Formula presented.) copies of (Formula presented.), and joining each and every vertex of the (Formula presented.) copy of (Formula presented.) with the (Formula presented.) vertex of (Formula presented.) by an edge. In this paper, for path (Formula presented.) and cycle (Formula presented.), (Formula presented.), we determine (Formula presented.) when (Formula presented.) is even, and give an upper bound for the same when (Formula presented.) is odd. Also, for (Formula presented.), we determine the radio number of (Formula presented.) when (Formula presented.) is even, and give both upper and lower bounds for (Formula presented.) when (Formula presented.) is odd. © 2018 Kalasalingam University. Published with license by Taylor & Francis Group, LLC.