Metric spaces in chess and international chess pieces graph diameters

This paper aims to study the graph radii and diameters induced by the k-dimensional versions of the well-known six international chess pieces on every finite {n×n×· · ·×n} ⊆ ℤ^k lattice since they originate as many interesting metric spaces for every proper pair (n, k). For this purpose, we finally discuss a mathematically consistent generalization of all the planar FIDE chess pieces to an appropriate k-dimensional environment, finding (for all k ∈ ℤ⁺) the exact values of the graph radii and diameters of the k-rook, k-king, k-bishop, and the corresponding values for the 3-queen, 3-knight, and 3-pawn. We also provide tight bounds for the graph radii and diameters of the k-queen, k-knight, and k-pawn, holding for every k ⩾ 4.