The Menger number of the Cartesian product of graphs

In a real-time system, the Menger number ζ l ( G ) is an important measure of the communication efficiency and fault tolerance of the system G . In this paper, we obtain a lower bound for the Cartesian product graph. We show that ζ l 1 + l 2 ( G 1 × G 2 ) ≥ ζ l 1 ( G 1 ) + ζ l 2 ( G 2 ) for l 1 ≥ 2 and l 2 ≥ 2 . As an application of the result, we determine the exact values ζ l ( G ) of the grid network G = G ( m 1 , m 2 , … , m n ) for m i ≥ 2 ( 1 ≤ i ≤ n ) and l ≥ ∑ i = 1 n m i . This example shows that the lower bound of ζ l 1 + l 2 ( G 1 × G 2 ) obtained is tight.