The Radio Number for Some Classes of the Cartesian Products of Complete Graphs and Cycles

Abstract

A radio coloring of graphs is a modification of the frequency assignment problem. For a connected simple graph G, a mapping g of the vertices of G to the positive integers (colors) such that for every pair u and v of G, | g(u) − g(v)| is at least 1 + diam(G) − d(u, v), is called a radio coloring of G. The largest color used by g is called span of g, denoted by rn(g). The radio number, rn(G), is the least of { rn(g) : g is a radio coloring of G }. In this paper, for n ⩾ 7 we obtain the radio number of Cartesian product of complete graph K n and cycle C m , K n ☐C m , for n even and m odd, and for n odd and m 5 (mod 8).