Two kinds of conditional connectivity of hypercubes

A subset F subset of V(G) is called an h-extra r-component cut of G if G - F is disconnected and there are at least r components, each component has at least h + 1 vertices. The cardinality of a minimum h-extra r-component cut of G, denoted by c kappa(h)(r)(G), is the h-extra r-component connectivity of G. In this paper, we introduce a novel connectivity called the g-good r-component connectivity. For F subset of V(G), if G - F is disconnected and there are at least r components and each vertex v is an element of G - F has at least g neighbors, then F is called a g-good r-component cut of G; the g-good r-component connectivity of G, denoted by ac C kappa(g,r)(G), is the minimum cardinality of a g-good r-component cut of G. In this work, we prove that C kappa(2)(3)(Q(n)) = 6n - 20 for n >= 9 and C kappa(2,3)(Q(n)) = 8n - 24 for n >= 11, where Q(n) is n-dimension hypercube.