Multiplicative graphs and semi-lattice endomorphisms in the category of graphs

A graph K is called multiplicative if whenever a categorical product of two graphs admits a homomorphism to K, then one of the factors also admits a homomorphism to K. We prove that all circular graphs K k / d such that k / d < 4 are multiplicative. This is done using semi-lattice endomorphism in (the skeleton of) the category of graphs to prove the multiplicativity of some graphs using the known multiplicativity of the odd cycles.