Abstract: We considered the r-hued chromatic number of Cartesian product of complete graph K_n and complete bipartite graph K_n□K_m,s. Firstly, we classified the positive integer r according to its different values, and combined with the properties of K_n□K_m,s, we characterized a lower bound of the r-hued chromatic number of the graph. Secondly, we found a specific (k,r)-coloring of K_n□K_m,s, so we characterized an upper bound of the r-hued chromatic number of the graph. Finally, we determined the r-hued chromatic number of K_n□K_m,s.

Key words: (k,r)-coloring, r-hued chromatic number, Cartesian product of graph