Optimal $L(d,1)$-labeling of certain direct graph bundles cycles over cycles and Cartesian graph bundles cycles over cycles

An $L(d,1)$-labeling of a graph G is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least d and those at a distance of two receive labels that differ by at least one, where d≥ 1. Let λᵈ₁ (G) denote the least λ such that G admits an $L(d,1)$-labeling using labels from \0,1,… , λ \. We prove that λᵈ₁(X)≤ 2d+2 for certain direct graph bundle X= Cₘ×σ_ Cₙ and certain Cartesian graph bundle X= Cₘσ_ Cₙ, where σ_ is a cyclic -shift, with equality if 1≤ d≤ 4.