Minimal orientations of colour critical graphs (Q1894707)

An orientation of a graph \(G\) is called minimal if it possesses precisely one directed path of length \(\chi(G)- 1\). \textit{T. Gallai} [On directed paths and circuits, Theory of graphs, Proc. Colloquium Tihany, Hungary 1966, 115-118 (1968; Zbl 0159.544)] asked whether every colour critical graph has a minimal orientation. The author constructs a counterexample exhibiting a critically 4-chromatic graph on 14 vertices which has no minimal orientation. The author also proves that the answer is affirmative for \(k\)-critical graphs whose maximal degree is at most \(k\) when \(k\geq 5\).