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The (2 1)-labeling of a graph is an abstraction of the problem of assigning (integer) frequencies to radio transmitters, such that transmitters that are "close", receive different frequencies, and those that are "very close" receive frequencies that are further apart. The least span of frequencies in such a labeling is referred to as the -number of the graph. Let be odd 5, = ( 3) 2 and let each be a multiple of . It is shown that ( ) is equal to the theoretical minimum of 1, where denotes a cycle of length and " " denotes the Cartesian product of graphs. The scheme works for a vertex partition of into smallest (independent) dominating sets.

The matrix S in Theorem 3 is obtained as S = 3 4 01 01 3 4Since the system equations discussed in [15] require c i = 1(i = 1; 2), the condition given in [15] is not applied. In addition, we can also see that S is not a nonsingular M matrix, so the condition given in [15] does not also hold for the matrices given above. III. CONCLUSIONSA set of criteria have been derived ensuring the global asymptotic stability of delayed cellular neural networks (DCNNs) with more general output functions by introducing ingeniously infinitely real parameters, constructing suitable Lyapunov functionals and applying some analysis techniques, and these criteria are independent of delays and possess infinitely adjustable real parameters w i > 0, 3 ij , 3 ij , 3 ij , 3 ij , ij, ij, ij, ij are any real numbers with 3 ij + 3 ij = 1, ij + ij = 1, 3 ij + 3 ij = 1, ij + j = 1 (i; j = 1; 2; . . . ; n). These are of prime importance and great interest in many application fields and the design of networks. In addition, the methods of this paper may extended to discuss more complicated systems such as Hopfield neural networks(HNNs) and bi-directional associative memory (BAM) networks. ACKNOWLEDGMENTThe author would like to thank the referees and the associate editor for suggesting improvements in presentation.

Weichsel (Proc. Amer. Math. Soc. 13 (1962) 47-52) proved that the Kronecker product of two connected bipartite graphs consists of two connected components. A condition on the factor graphs is presented which ensures that such components are isomorphic. It is demonstrated that several familiar and easily constructible graphs are amenable to that condition. A partial converse is proved for the above condition and it is conjectured that the converse is true in general.

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