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

Abstract

A labelling of a graph G is a mapping $\pi :S \rightarrow {\mathcal {L}}$, where ${\mathcal {L}}\subset {\mathbb {R}}$ and $S \subseteq V(G)\cup E(G)$. If $S=E(G)$, $\pi$ is an ${\mathcal {L}}$-edge-labelling and, if $S=V(G)\cup E(G)$, $\pi$ is an ${\mathcal {L}}$-total-labelling. For each $v\in V(G)$, the colour of v under $\pi$ is defined as $C_{\pi }(v) = \sum _{uv \in E(G)}{\pi (uv)}$ if $\pi$ is an ${\mathcal {L}}$-edge-labelling; and $C_{\pi }(v) = \pi (v)+\sum _{uv \in E(G)}{\pi (uv)}$ if $\pi$ is an ${\mathcal {L}}$-total-labelling. Labelling $\pi$ is a neighbour-distinguishing ${\mathcal {L}}$-edge-labelling (neighbour-distinguishing ${\mathcal {L}}$-total-labelling) if $\pi$ is an ${\mathcal {L}}$-edge-labelling (${\mathcal {L}}$-total-labelling) and $C_{\pi }(u)\ne C_{\pi }(v)$, for every edge $uv \in E(G)$. 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 $\{1,2,3\}$-edge-labelling. In 2010, Przybyło and Woźniak posed the 1,2-Conjecture, which states that every simple graph has a neighbour-distinguishing $\{1,2\}$-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 $\{t,2t\}$-total-labellings and neighbour-distinguishing $\{t,2t,3t\}$-edge-labellings, for $t\in {\mathbb {R}}\backslash \{0\}$; and we prove that all powers of cycles have neighbour-distinguishing $\{a,b\}$-total-labellings, and neighbour-distinguishing $\{t,2t,3t\}$-edge-labellings, for $a,b,t\in {\mathbb {R}}$, $a\ne b$ and $t\ne 0$