On the (p,1)-total number of near-ladders and generalized Petersen graphs - Omai, M. M.; Campos, C. N.; Luiz, A. G. (Proceedings of the Theory of Computation Meeting (ETC), 2022)
A k-(p, 1)-total labelling of a simple graph G is a function π: V(G)∪E(G) → {0, . . . , k} such that: |π(uv) − π(u)| ≥ p and |π(uv) − π(v)| ≥ p for uv ∈ E(G); π(uv) ≠ π(vw) for uv, vw ∈ E(G); and π(u) ≠ π(v) for uv ∈ E(G). The least integer k for which G admits a k-(p, 1)-total labelling is denoted λ_p^t(G). In this work, we show that: λ_p^t(G) = p + 4, p ≥ 3, for non-bipartite near-ladder graphs and generalized Petersen graphs P(ℓ, 2), ℓ ≥ 6; and λ_2^t(G) = 5 for graphs P(ℓ, 2), ℓ ≥ 6.