On α-total domination in prisms and Mobius ladders

Let G = (V, E) be a graph with no isolated vertex. A subset of vertices S is a total dominating set if every vertex of G is adjacent to some vertex of S. For some a with 0 < alpha <= 1, a total dominating set S in G is an a-total dominating set if for every vertex v is an element of V \ S, vertical bar N(v) boolean AND S vertical bar >= alpha vertical bar N(v)vertical bar. The alpha-total domination number of G, denoted by gamma(alpha t)(G), is the minimum cardinality of an alpha-total dominating set of G. In [1], Henning and Rad posed the following question: Let G be a connected cubic graph with order n. Is it true that gamma(alpha t)(G) <= (2n)(3) for 1/3 < (2)(3) and gamma(alpha t)(G) <= (3n)(4) for (2)(3) < alpha <= 1 ? In this paper, we find alpha-total domination numbers for two classes of connected cubic graphs, namely the prisms and the M5bius ladders. All the exactly values are less than the hounds in the question. We give a positive answer toward this question on these two classes of connected cubic graphs.