Acyclic orientations of a graph and chromatic and independence numbers

Abstract

The determinations of chromatic and independence numbers of a graph are represented as problems in optimization over the set of acyclic orientations of the graph. Specifically χ = min ω∈Ωk ω and β 0 = max ω∈Ωk ω where χ is the chromatic number, β 0 is the independence number, Ω is the set of acyclic orientations, l ω is the length of a maximum chain, and k ω is the cardinality of a minimum chain decomposition. It is shown that Dilworth's theorem is a special case of the second equality.