The existence problem for colour critical linear hypergraphs

Abstract

One of the most extensively studied concepts of ordinary graph theory is that of chromatic number. We remind the reader that a 2-graph is said to have chromatic number r if r is the least positive integer for which there exists some way of colouring the vertices of the graph in r colours so that no edge joins vertices of the same colour. This notion generalizes in a natural way to hypergraphs. A hypergraph is an ordered pair (V, ~ ) where V is a finite non-empty set and is a non-empty collection of subsets of V. We shall always assume that V= tJ ~-, so that we may speak of the hypergraph .~ rather than (V, ~-). The elements of [J .~ are called the vertices of ~while the elements of ~are called the edges of J . A hypergraph ~ is said to be uniform if for all F ~ , IF]=n for some integer n~2 . Then ~is called an n-graph. An n-graph ~is said to be linear if [F(~F'I _2 for all F ~ . We call c# an r-colouring of g . ~is r-chromatic if it is r-colourable but not (r-1)-colourable and we then call r the chromatic number of ~ . ~ is r-critical if it is r-chromatic and all its proper subgraphs are ( r 1)-colourable. ~ is critical if it is r-critical for some r. Critical 2-graphs were first investigated by Dirac and subsequently by many other authors (see for example [9] and references given there). It is a simple matter to verify that the only 3-critical 2-graphs are the circuits of odd length. It was hoped at one time that a characterization of the 4-critical 2-graphs would be useful in tackling the celebrated Four Colour Problem. However, no such characterization has been found and, in fact, recent results (see, for example, SIMONOVITS [11] and TOFT [12]) indicate that the 4-critical 2-graphs may be quite complicated and that perhaps no simple characterization is possible. By an (m, n, r)-graph we shall mean an r-critical n-graph on m vertices. There are two questions which arise: (A) Given integers n and r, for which integers m do (m, n, r)-graphs exist? (B) Given integers n and r, for which integers m do linear (m, n, r)-graphs exist? For 2-graphs, and here the two problems coincide, (m, 2, 3)-graphs exist only when m ~ 3 is odd. This is just a restatement of the fact that the only 3-critical 2-graphs are the odd circuits. For r ~ 4 , DmAC [4] proved that (m, 2, r)-graphs exist only when m=r or m>=r+2. For n~3 , let M(n, r) = ( n 1 ) ( r 1 ) + l .