The concept of $[r,s,t]$-colourings was recently introduced by Hackmann, Kemnitz and Marangio [3] as follows: Given non-negative integers $r,s$ and $t$, an $[r,s,t]$-colouring of a graph $G=(V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the colour set $\{1,2, \ldots, k\}$ such that $|c(v_i)-c(v_j)| \geq s$ for every two adjacent vertices $v_i$, $v_j$, $|c(e_i)-c(e_j)| \geq s$ for every two adjacent edges $e_i$, $e_j$, and $|c(v_i)-c(e_j)| \geq t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$-chromatic number $\chi_{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an [r; s; t]-colouring. In this paper, we determine the $[r,s,t]$-chromatic number for paths.
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