Some properties of non-bicolorable hypergraphs and the four-color problem Original Research Article

The hypergraphs whose chromatic number is at most 2 (“bicolorable” hypergraphs) were introduced by Miller (1937) under the name of “set-systems with Property B”. This concept appears in Number Theory (see V. Chvatal (1971) and R.L. Graham et al. (1986)). It is also useful for some problems in positional games and Operations Research (see C. Berge (1992). (1990) and P. Erdös and J.L. Selfridge (1973)); different results have been found under the form of inequalities involving the sizes of the edges, the number of vertices, etc. (see P. Erdös and L. Lovász (1975), P. Hansen and M. Lorea (1978) and M. Herzog and J. Schönheim (1972)). 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 (C. Berge (1989). (1992) and P.D. Seymour (1974)). In this paper, instead of edge-critical hypergraphs, we study the vertex-critical hypergraphs; the applications are more numerous, and it appears that somewhat stronger results could imply the famous “four-color theorem”