Distance-two labellings of Hamming graphs

Let j ≥ k ≥ 0 be integers. An `-L(j, k)-labelling of a graph G = (V , E) is a mapping φ : V → {0, 1, 2, . . . , `} such that |φ(u)−φ(v)| ≥ j if u, v are adjacent and |φ(u)−φ(v)| ≥ k if they are distance two apart. Let λj,k(G) be the smallest integer ` such that G admits an `-L(j, k)-labelling. Define λj,k(G) to be the smallest ` if G admits an `-L(j, k)-labelling with φ(V ) = {0, 1, 2, . . . , `} and∞ otherwise. An `-cyclic L(j, k)-labelling is a mapping φ : V → Z` such that |φ(u) − φ(v)|` ≥ j if u, v are adjacent and |φ(u) − φ(v)|` ≥ k if they are distance two apart, where |x|` = min{x, ` − x} for x between 0 and `. Let σj,k(G) be the smallest ` − 1 of such a labelling, and define σ j,k(G) similarly to λj,k(G). We determine λ2,0, λ2,0, σ2,0 and σ 2,0 for all Hamming graphs Kq1 Kq2 · · · Kqd (d ≥ 2, q1 ≥ q2 ≥ · · · ≥ qd ≥ 2) and give optimal labellings, with the only exception being 2q ≤ σ 2,0(Kq Kq) ≤ 2q+1 for q ≥ 4. We also prove the following ‘‘sandwich theorem’’: If q1 is sufficiently large then λ2,1(G) = λ2,1(G) = σ 2,1(G) = σ2,1(G) = λ1,1(G) = λ1,1(G) = σ 1,1(G) = σ1,1(G) = q1q2−1 for any graphG between Kq1 Kq2 and Kq1 Kq2 · · · Kqd , and moreover we give a labelling which is optimal for these eight invariants simultaneously. © 2009 Elsevier B.V. All rights reserved.