The colour theorems of Brooks and Gallai extended

One of the basic results in graph colouring is Brooksʹ theorem [4] which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum degree. As an extension of this result, Gallai [6] characterized the subgraphs of k-colour-critical graphs induced by the set of all vertices of degree k − 1. The choosability version of Brooksʹ theorem was proved, independently, by Vizing [9] and by Erdös et al. [5]. As Thomassen pointed out in his talk at the Graph Theory Conference held at Oberwolfach, July 1994, one can also prove a choosability version of Gallaiʹs result. All these theorems can be easily derived from a result of Borodin [2, 3] and Erdös et al. [5] which enables a characterization of connected graphs G admitting a color scheme L such that |L(x)| ⩾ dG (x) for all x ϵ V (G) and there is no L-colouring of G. In this note, we use a reduction idea in order to give a new short proof of this result and to extend it to hypergraphs