Acyclic orientations of a graph and chromatic and independence numbers

AbstractThe 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 = minω∈Ωkω and β0 = maxω∈Ωkω where χ is the chromatic number, β0 is the independence number, Ω is the set of acyclic orientations, lω is the length of a maximum chain, and kω is the cardinality of a minimum chain decomposition. It is shown that Dilworth's theorem is a special case of the second equality