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 $V\left(G\right)$ 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 $d\ge 1$. The span of f is the difference between the largest and the smallest numbers in $f\left(V\right)$. The ${\lambda }_1^d$-number of G, denoted by ${\lambda }_1^d\left(G\right)$, is the minimum span over all $L(d,1)$-labelings of G. We prove that ${\lambda }_1^d\left(X\right)\le 2d+2$, with equality if $1\le d\le 4$, for direct graph bundle $X=C_m{\times }^{{\sigma }_{\ell }}{C}_n$ and Cartesian graph bundle $X=C_m{\square }^{{\sigma }_{\ell }}{C}_n$, if certain conditions are imposed on the lengths of the cycles and on the cyclic ℓ-shift ${\sigma }_{\ell }$.