The generalized 3-connectivity of graph products

The generalized k-connectivity K-k(G) of a graph G, which was introduced by Chartrand et al. (1984), is a generalization of the concept of vertex connectivity. For this generalization, the generalized 2-connectivity K-2(G) of a graph G is exactly the connectivity K(G) of G. In this paper, let G be a connected graph of order n and let H be a 2-connected graph. For Cartesian product, we show that K-3(G square H) >= K-3(G) + 1 if K (G) = K-3 (G); K-3 (G square H) >= k(3)(G) + 2 if K(G) > K-3(G). Moreover, above bounds are sharp. As an example, we show that K-3 [GRAPHICS] = 2k - 1, where C-n1 is a cycle. For lexicographic product, we prove that K-3 (H circle G) >= max{3 delta(G) + 1, inverted right perpendicular 3n+1/2 inverted left perpendicular} if delta (G) < 2n-1/3, and K-3 (H circle G) = 2n if delta (G) >= 2n-1/3. (C) 2016 Elsevier Inc. All rights reserved.