No-hole k-tuple (r + 1)-distant colorings Original Research Article

We study the existence and efficiency of the no-hole k-tuple (r + 1)-distant colorings of a graph, or Nkr-colorings. More specifically, we want to assign a set of k positive integers (the k-tuple requirement) to each vertex of a graph such that adjacent vertices must get numbers at least (r + 1) apart (the (r + 1)-distant requirement), and in addition, the set of all used numbers is a consecutive set (the no-hole requirement). The case k = 1 was studied by Roberts (1993) and Sakai and Wang (1993).