Let \(\tau(G)\) denote the number of vertices in a longest path of a graph \(G\). The \(n\)th detour number \(\chi_n(G)\) of a graph \(G\) is the minimum number of colours required to colour the vertices of \(G\) such that no path with more than \(n\) vertices is monocoloured. It is shown that the path partition conjecture, formulated by P. Mihók (see \textit{M. Borowiecki}, et al. [Discuss. Math., Graph Theory 17, 5-50 (1997; Zbl 0902.05026)]), is strongly related to the conjecture that \(\chi_n(G)\leq \lceil \frac{\tau(G)}n \rceil\) for every graph \(G\) and every \(n\geq 1\). Some results determining \(\chi_n(G)\) for particular classes of graphs are presented. Sufficient conditions for graphs with \(n\)th chromatic number at most \(2\) are provided as well.