Two remarks on retracts of graph products

AbstractLet H be a bipartie graph and let Gn be the Mycielski graph with χ(G) = n, n ⩾ 4. Then the chromatic number of the strong product of Gn by H is at most 2n − 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