No-hole ( r+1)-distant colorings

Abstract

We study a generalization of the no-hole 2-distant colorings introduced by Roberts (1990). Here we want to color the vertices of a graph with consecutive positive integers so that adjacent vertices get colors which differ by more than a fixed positive integer r⩾1. (When r=1 we get a no-hole 2-distant coloring.) Surprisingly, this concept turns out to be closely related to the Hamiltonian path problem. Besides examining the existence of such generalized colorings, we also seek to find one which uses a relatively small number of consecutive colors. We settle two questions left open in Roberts (1990).