Two remarks on retracts of graph products

Abstract Let H be a bipartie graph and let G n be the Mycielski graph with χ( G ) = n , n ⩾ 4. Then the chromatic number of the strong product of G n by H is at most 2 n − 2. We use this result to show that there exist strong products of graphs in which a projection of a retract onto a factor is not a retract of the factor. We also show that in the Cartesian product of graphs G and H , any tringles of G transfer in H , whenever G and H are connected and G is strongly-triangulated, weakly-triangulated or four-cycle free.