Neighbour-distinguishing labellings of powers of paths and powers of cycles

A labelling of a graph G is a mapping $$\pi :S \rightarrow {\mathcal {L}}$$π:S→L, where $${\mathcal {L}}\subset {\mathbb {R}}$$L⊂R and $$S \subseteq V(G)\cup E(G)$$S⊆V(G)∪E(G). If $$S=E(G)$$S=E(G), $$\pi $$π is an $${\mathcal {L}}$$L-edge-labelling and, if $$S=V(G)\cup E(G)$$S=V(G)∪E(G), $$\pi $$π is an $${\mathcal {L}}$$L-total-labelling. For each $$v\in V(G)$$v∈V(G), the colour of v under $$\pi $$π is defined as $$C_{\pi }(v) = \sum _{uv \in E(G)}{\pi (uv)}$$Cπ(v)=∑uv∈E(G)π(uv) if $$\pi $$π is an $${\mathcal {L}}$$L-edge-labelling; and $$C_{\pi }(v) = \pi (v)+\sum _{uv \in E(G)}{\pi (uv)}$$Cπ(v)=π(v)+∑uv∈E(G)π(uv) if $$\pi $$π is an $${\mathcal {L}}$$L-total-labelling. Labelling $$\pi $$π is a neighbour-distinguishing $${\mathcal {L}}$$L-edge-labelling (neighbour-distinguishing $${\mathcal {L}}$$L-total-labelling) if $$\pi $$π is an $${\mathcal {L}}$$L-edge-labelling ($${\mathcal {L}}$$L-total-labelling) and $$C_{\pi }(u)\ne C_{\pi }(v)$$Cπ(u)≠Cπ(v), for every edge $$uv \in E(G)$$uv∈E(G). In 2004, Karónski, Łuczac and Thomasson posed the 1,2,3-Conjecture, which states that every simple graph with no isolated edge has a neighbour-distinguishing $$\{1,2,3\}$${1,2,3}-edge-labelling. In 2010, Przybyło and Woźniak posed the 1,2-Conjecture, which states that every simple graph has a neighbour-distinguishing $$\{1,2\}$${1,2}-total-labelling. In this work, we contribute to the study of these conjectures by verifying the 1,2,3-Conjecture and 1,2-Conjecture for powers of paths and powers of cycles. We also obtain generalizations of these results: we prove that all powers of paths have neighbour-distinguishing $$\{t,2t\}$${t,2t}-total-labellings and neighbour-distinguishing $$\{t,2t,3t\}$${t,2t,3t}-edge-labellings, for $$t\in {\mathbb {R}}\backslash \{0\}$$t∈R\{0}; and we prove that all powers of cycles have neighbour-distinguishing $$\{a,b\}$${a,b}-total-labellings, and neighbour-distinguishing $$\{t,2t,3t\}$${t,2t,3t}-edge-labellings, for $$a,b,t\in {\mathbb {R}}$$a,b,t∈R, $$a\ne b$$a≠b and $$t\ne 0$$t≠0