Constructing a large class of supertough graphs

For each odd r ≥ 3andeachn of the form 2kb(r +2 − 2b) fork ≥ 1 and 1 ≤ b ≤ (r − 1)/2, the first author has constructed an r-regular r/2-tough graph on n vertices. In this paper, we provide an alternate and more advantageous construction. First, both our new construction and its proof are simpler. Second, we use an extension of the notion of graph inflations, as used in a construction given by Chvátal in the case that r = 3. Third, we generalize a construction of our own used in the case that r = 5, that allowed us to further remove the restrictions on the number of vertices n