About vertex-critical non-bicolorable hypergraphs

Claude BERGE C.N.R.S., Paris 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 could imply the famous "four-color theorem". I Vertex-critical hypergraphs and the four-color problem Let H = (E1> E2, ... , Ern) be a hypergraph which is simple (i.e. Ei:2 Ej implies i = j ). Denote by X = {x b X2,"" xn } its vertex-set, and for A ~ X, denote by HI A the partial hypergraph HI A = ( E / EEH , E ~ A) (this family can be empty). We denote also by H-H(Xi) the hypergraph obtained from H by removing all the edges which contain the vertex xi (and all the vertices which become of degree 0 ). Let X(H) denote the chromatic number of H, i.e. the least number of colors needed to color the vertices so that no edge is monochromatic (except, of course, the edges of cardinality one, or "loops"). The hypergraph H is edge-critical (with respect to the non-bicolorability) if X(H) > 2 and Australasian Journal of Combinatorics ~(1994), pp.211-219 H-E is bicolorable for every EEH. A hypergraph H is vertex-critical if X(H) > 2 and' H-H(x) is bicolorable for every vertex x. Clearly, every hypergraph which is not bicolorable has a partial hypergraph which is edge-critical, and every edge-critical hypergraph is also vertex-critical. Furthermore, every which is not bicolorable contains a set A of vertices such that the hypergraph HI A is vertex-critical. Some classical examples of edge-critical hypergraphs are: the finite projective plane with 7 points, the complete r-uniform hypergraph ~r-l of order 2r-1, the Lovasz hypergraph Lr , the complement of L3, etc ... (see [2], Chap.2). Seymour [14] has characterized the edge-critical hypergraphs having as many vertices as edges (by association with strongly connected directed graphs without even circuits). Number Theory provides several examples of vertex-critical hypergraphs which are not edge-critical: Consider the "triangle T hypergraph" Kn ~ that is the hypergraph whose vertices are the edges of the complete graph Kn and whose edges are the triangles of Since the T Ramsey number R(3,3) is 6, we have X(K6 ) = 3 and ) = 2. The hypergraph K~ is vertex-critical: if the vertices of K6 are a,b,c,d,e,f, and if the edge af is removed. the other can be colored with two colors without producing a monochromatic triangle ( for with blue: ab, bc , hf ,ae, ed; ef, cd; with red: ac, ad, bd, be, ce, cf, df). Nevertheless, it is easy to check that the hypergraph K~ is not edge-critical. The well known theorem of van der Waerden ("If the natural numbers are split into lYvo classes, then for every k at least one class contains an arithmetic progression of k terms, fl) can be as follows: If Ak is a finite set of integers such that in every bicoloring of at least one color class contains an arithmetic progression of k terms, and if is minimal, then the arithmetic progressions of k terms define a vertex-critical hypergraph (which is not necessarly edge-critical).