The Level of Nonmultiplicativity of Graphs

We introduce a parameter called the of a graph, which is related to Hedetniemi's conjecture. We show that this parameter is equal to the number of factors in a factorization of the graph into a product of multiplicative graphs. Apart from the known multiplicative graphs, no graph is known to have a finite level of nonmultiplicativity. We show that the countably infinite complete graph has an infinite level of nonmultiplicativity and that there exist Kneser graphs with arbitrarily high levels of nonmultiplicativity