Largest 6-regular toroidal graphs for a given diameter

We show that a 6-regular graph of diameter d embedded on the torus can have at most 3d + 3d+ 1 vertices having exhibited a graph of this order for each d ≥ 1. 1 6-regular Toroidal Graphs In [1] it is proven that all 6-regular toroidal graphs can have their vertices arranged in a rectangular based grid structure with parallel diagonals across every sub-rectangle in the grid. A special case of this structure is for graphs we shall refer to as H(n, k). They are constructed by taking a cycle with n vertices consecutively labelled 0, . . . , n− 1 and adding edges from each vertex to the vertices with labels k and k + 1 (modulo n) around the cycle. We note that H(n, k) is vertex transitive and hence one can determine the diameter simply by calculating the eccentricity of one vertex. 2 A Graph of diameter d with order 3d2 + 3d + 1 We define the graph Hd := H(3d +3d+1, 3d+1) with the n := 3d+3d+1 vertices numbered consecutively clockwise from 0 to 3d + 3d. Note that this implies that vertex i also has edges from vertices n−(3d+2+i)≡ 3d2−1−i and n−(3d+1+i)≡ 3d − i. This implies that Hd is 6-regular since none of -1, 1, 3d+ 1, 3d+ 2, 3d − 1 and 3d are equal for any positive integer d. The edges between vertices with a label difference of 1 will be referred to as exterior edges and all other edges will be jump edges. Theorem 2.1 The diameter of Hd is exactly d.