Some properties of non-bicolorable hypergraphs and the four-color problem (Q1917261)

Hypergraphs whose chromatic number is at most 2 are called bicolorable hypergraphs. A non-bicolorable hypergraph which becomes bicolorable when any of its edges (vertices) is removed is called edge-critical (vertex-critical). Edge-critical bicolorable hypergraphs have widely been studied in the literature. In this paper, the author points out that the so far rarely studied vertex-critical hypergraphs have numerous applications, in particular, there is an intimate relationship to the four-colour theorem.