Neighbour-Distinguishing Labellings of Families of Graphs

Abstract

A labelling of a graph G is a mapping $\pi :S \rightarrow {\mathcal {L}}$, where ${\mathcal {L}}\subset {\mathbb {R}}$ and $S\in \{E(G), V(G)\cup E(G)\}$. 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. The pair $(\pi ,c_{\pi })$ 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 this work, we show that split graphs, regular cobipartite graphs, complete multipartite graphs and cubic graphs have neighbour-distinguishing $\{a,b,c\}$-edge-labellings, for distinct $a,b,c \in {\mathbb {R}}$ (in some cases $a,b,c \ge 0$). For split graphs and regular cobipartite graphs we also prove they admit neighbour-distinguishing $\{a,b\}$-total-labellings. Furthermore, we show that flower snarks and some subfamilies of split graphs and regular cobipartite graphs have neighbour-distinguishing $\{a,b\}$-edge-labellings and prove that some families of split graphs do not have neighbour-distinguishing ${\mathcal {L}}$-edge-labellings, for ${\mathcal {L}} = \{a,2a\}$ and ${\mathcal {L}}=\{0,a\}$, $a,b \in {\mathbb {R}}\backslash \{0\}$, $a \ne b$.