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

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).