Further results on independence in direct-product graphs - Jha, P. K. (Charles Babbage Res Ctr, 2000)
For a graph G, let alpha(G) and tau(G) denote the independence number of G and the matching number of G, respectively. Further, let G x H denote the direct product (also known as Kronecker product, cardinal product, tensor product., categorical product and graph conjunction) of graphs G and H. It is known that alpha(G x H) greater than or equal to max{alpha(G) . \H\, alpha(H) . \G\} =: alpha(G x H) and that tau(G x H) greater than or equal to 2 . tau(G) . tau(H) =: tau(G X H). It is shown that an equality/inequality between ct and ct is independent of an equality/inequality between tau and tau. Further, several results are presented on the existence of a complete matching in each of the two connected components of the direct product of two bipartite graphs. Additional results include an upper bound on alpha(G x H) that is achievable in certain cases.