Minimal non-extensible precolorings and implicit-relations

Abstract: 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 non-extensible 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.