Quantum Chromatic Of G×K_n And Quantum Multiplicative Graphs

In this paper,we use the quantum homomorphism of graph to study the categorical product of graph.By introducing the categorical product of the classical homomorphis-m into the quantum homomorphism,the quantum chromatic number of the categorical product of a class graphs is obtained.We introduce the concept of quantum multiplicative graph and show that K2 is quantum multiplicative graph.Moreover,we obtain a neces-sary and sufficient condition that a graph is quantum multiplicative.Finally,we construct graph exhibiting separations between clique number and quantum clique number.