Distance-two labellings of Hamming graphs

Let j ≥ k ≥ 0 be integers. An ℓ-L (j, k)-labelling of a graph G = (V, E) is a mapping φ{symbol} : V → {0, 1, 2, ..., ℓ} such that | φ{symbol} (u) - φ{symbol} (v) | ≥ j if u, v are adjacent and | φ{symbol} (u) - φ{symbol} (v) | ≥ k if they are distance two apart. Let λ_{j, k} (G) be the smallest integer ℓ such that G admits an ℓ-L (j, k)-labelling. Define over(λ, -)_{j, k} (G) to be the smallest ℓ if G admits an ℓ-L (j, k)-labelling with φ{symbol} (V) = {0, 1, 2, ..., ℓ} and ∞ otherwise. An ℓ-cyclic L (j, k)-labelling is a mapping φ{symbol} : V → Z_ℓ such that | φ{symbol} (u) - φ{symbol} (v) |_ℓ ≥ j if u, v are adjacent and | φ{symbol} (u) - φ{symbol} (v) |_ℓ ≥ k if they are distance two apart, where | x |_ℓ = min {x, ℓ - x} for x between 0 and ℓ. Let σ_{j, k} (G) be the smallest ℓ - 1 of such a labelling, and define over(σ, -)_{j, k} (G) similarly to over(λ, -)_{j, k} (G). We determine λ_{2, 0}, over(λ, -)_{2, 0}, σ_{2, 0} and over(σ, -)_{2, 0} for all Hamming graphs K_q1 □ K_q2 □ ⋯ □ K_qd (d ≥ 2, q₁ ≥ q₂ ≥ ⋯ ≥ q_d ≥ 2) and give optimal labellings, with the only exception being 2 q ≤ over(σ, -)_{2, 0} (K_q □ K_q) ≤ 2 q + 1 for q ≥ 4. We also prove the following "sandwich theorem": If q₁ is sufficiently large then λ_{2, 1} (G) = over(λ, -)_{2, 1} (G) = over(σ, -)_{2, 1} (G) = σ_{2, 1} (G) = λ_{1, 1} (G) = over(λ, -)_{1, 1} (G) = over(σ, -)_{1, 1} (G) = σ_{1, 1} (G) = q₁ q₂ - 1 for any graph G between K_q1 □ K_q2 and K_q1 □ K_q2 □ ⋯ □ K_qd, and moreover we give a labelling which is optimal for these eight invariants simultaneously.