On the complexity of graph coloring with additional local conditions

Let G = (V, E) be a finite simple graph. Recall that a proper coloring of G is a mapping phi : V -> {1, ..., k} such that every color class induces an independent set. Such a phi is called a semi-matching coloring if the union of any two consecutive color classes induces a matching. We show that the semi-matching coloring problem is NP-complete for any fixed k >= 3, and we get the same result for another version of this problem in which any triangle of G is required to have vertices whose colors differ at least by three. (C) 2018 Elsevier B.V. All rights reserved.