Sharper Lower Bounds in the Maximum Degree and Diameter Bounded Subgraph Problem in the Mesh

The Maximum Degree and Diameter Bounded Subgraph Problem (MaxDDBS) asks: given a host graph G, a bound on maximum degree \Delta, and a diameter D, what is the largest subgraph of the host graph with degree bounded by \Delta and diameter bounded by D? In this paper, we investigate this problem when the host graph is the k-dimensional mesh. We provide lower bounds for the size of the largest subgraph of the mesh satisfying MaxDDBS for all k and \Delta > 3 that agree with the known upper bounds up to the first two terms, and show that for \Delta = 3, the lower bounds are at least the same order of growth as the upper bounds.