On the Basis and Chromatic Number of a Graph

Extract

The basis theorem for directed graphs is, in effect, a result on weakly ordered sets, and, in §1, a proof is given, based on Zorn's lemma, that generalizes, and perhaps clarifies the exposition in (1, Chapter 2). In §2, a graph G* is defined, on an arbitrary collection Q of non-void subsets of a set X (which includes all its one-element subsets), in such a way that the partitions of X into Q-sets correspond to the kernels of G*. Applied to the collection Q of non-null internally stable subsets of a graph G without loops, this identifies the chromatic number of G with the least cardinal number of any kernel of G*.