Coloring Rings

A ring is a graph whose vertex set can be partitioned into nonempty sets, , such that for all , the set can be ordered as so that . A hyperhole is a ring such that for all , is complete to . In this paper, we prove that the chromatic number of a ring is equal to the maximum chromatic number of a hyperhole in . Using this result, we give a polynomial-time coloring algorithm for rings. Rings formed one of the basic classes in a decomposition theorem for a class of graphs studied by Boncompagni et al [J. Graph Theory 91 (2019), 192–246.]. Using our coloring algorithm for rings, we show that graphs in this larger class can also be colored in polynomial time. Furthermore, we find the optimal -bounding function for this larger class of graphs, and we also verify Hadwiger's conjecture for it.