Coloring the square of the Cartesian product of two cycles

The square G 2 of a graph G is defined on the vertex set of G in such a way that distinct vertices with distance at most 2 in G are joined by an edge. We study the chromatic number of the square of the Cartesian product C m □ C n of two cycles and show that the value of this parameter is at most 7 except when ( m , n ) is ( 3 , 3 ) , in which case the value is 9, and when ( m , n ) is ( 4 , 4 ) or ( 3 , 5 ) , in which case the value is 8. Moreover, we conjecture that whenever G = C m □ C n , the chromatic number of G 2 equals ⌈ m n / α ( G 2 ) ⌉ , where α ( G 2 ) denotes the maximum size of an independent set in G 2 .