Minimal non-extensible precolorings and implicit-relations

In this paper I study a variant of the general vertex coloring problem called precoloring. Specifically, I study graph precolorings, by developing new theory, for characterizing the minimal non-extensible precolorings. It is interesting per se that, for graphs of arbitrarily large chromatic number, the minimal number of colored vertices, in a nonextensible precoloring, remains constant; only two vertices u, v suffice. Here, the relation between such u, v is called an implicit-relation, distinguishing two cases: (i) implicit-edges where u, v are precolored with the same color and (ii) implicit-identities where u, v are precolored distinct. Mathematics Subject Classification (2010). Primary 05C15, 05C75 and Secondary 05C90, 05C69.