About vertex-critical non-bicolorable hypergraphs.
Claude Berge · 1994
The hypergraphs whose chromatic number is ~ 2 ("bicolorable " hypergraphs) were introduced by E.W. Miller [13] under the name of "set-systems with Property B". This concept appears in Number Theory (see [5], [10]). It is also useful for some problems in positional games and Operations Research (see [3], [4], [7]); different results have been found under the form of inequalities involving the sizes of the edges, the number of vertices, etc... ( see [6], [11], [12]). A non-bicolorable hypergraph which becomes bicolorable when any of its edges is removed is called "edge-critical", and several of its properties can be found in the literature ([2], [4], [14]). In this paper, instead of edge-critical hypergraphs, we study the vertex-critical hypergraphs; the applications are more numerous, and it seems that somewhat stronger results