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 = ( V , E ) is a function f from the vertex set V ( G ) to the set of nonnegative integers such that the labels on adjacent vertices differ by at least d and the labels on vertices at distance two differ by at least one, where d ≥ 1 . The span of f is the difference between the largest and the smallest numbers in f ( V ) . The λ 1 d -number of G , denoted by λ 1 d ( G ) , is the minimum span over all L ( d , 1 ) -labelings of G . We prove that λ 1 d ( X ) ≤ 2 d + 2 , with equality if 1 ≤ d ≤ 4 , for direct graph bundle X = C m × σ ℓ C n and Cartesian graph bundle X = C m □ σ ℓ C n , if certain conditions are imposed on the lengths of the cycles and on the cyclic ℓ -shift σ ℓ .