[r, s, t]-colourings of paths

Abstract

The concept of $[r,s,t]$-colourings was recently introduced by Hackmann, Kemnitz and Marangio [A. Kemnitz, M. Marangio, $[r,s,t]$-Colorings of Graphs, Discrete Math., to appear] 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 $|c(v_i)-c(v_j)| \geq r$ 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.