Path 3-(edge-)connectivity of lexicographic product graphs

Dirac showed that in a (k - 1)-connected graph there is a path through all the k vertices. The path k-connectivity pi(k)(G) of a graph G, which is a generalization of Dirac's notion, was introduced by Hager in 1986. As a natural counterpart of path k-connectivity, the concept of path k-edge-connectivity omega(k)(G) of a graph G was introduced. Denote by H omicron G the lexicographic product of two graphs H and G. In this paper, for a 2-connected graph G and a graph H, we show pi(3)(G omicron H) >= vertical bar V(H)vertical bar and omega(3)(G omicron H) >= 3[vertical bar V(H)vertical bar/2] + r, where r = vertical bar V(H)vertical bar(mod2). Moreover, the bound is sharp. In addition, the upper bounds for path 3-(edge-)connectivity of the lexicographic product of a connected graph and some specific graphs are obtained. (C) 2019 Elsevier B.V. All rights reserved.