A note on the chromatic number of the square of the Cartesian product of two cycles

The square G 2 of a graph G is obtained from G by adding edges joining all pairs of nodes at distance 2 in G . In this note we prove that χ ( ( C m □ C n ) 2 ) ≤ 6 for m , n ≥ 40 . This confirms Conjecture 19 stated in [É. Sopena, J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327–2333].