Labelling Cayley Graphs on Abelian Groups

For given integers j ≥ k ≥ 1, an L(j, k)-labelling of a graph Γ is an assignment of labels—nonnegative integers—to the vertices of Γ such that adjacent vertices receive labels that differ by at least j, and vertices distance two apart receive labels that differ by at least k. The span of such a labelling is the difference between the largest and the smallest labels used, and the minimum span over all L(j, k)-labellings of Γ is denoted by λj,k(Γ). The minimum number of labels needed in an L(j, k)-labelling of Γ is independent of j and k, and is denoted by μ(Γ). In this paper we introduce a general approach to L(j, k)-labelling Cayley graphs Γ over Abelian groups and deriving upper bounds for λj,k(Γ) and μ(Γ). Using this approach we obtain upper bounds on λj,k(Γ) and μ(Γ) for graphs Γ admitting a vertex-transitive Abelian group of automorphisms. Hypercubes Qd are examples of such graphs, and as consequences we obtain upper bounds for λj,k(Qd) and μ(Qd). We also obtain the exact values of λj,k(Γ) (2k ≥ j ≥ k) and μ(Γ) for some Hamming graphs Γ. The result shows that, under certain arithmetic conditions, these two invariants rely only on k and the orders of the two largest complete graph factors of the Hamming graph.