Coloring of graphs with cost-functions whose marginal costs decrease
Wayne Goddardi, Deirdre LaVey, Morgan Schalizki · Ars Combinatoria · 2025
We consider a vertex-coloring problem where the amount one pays for using a color is a function of how many times the color is used. For a cost-function \(f\), we define the \(f\)-chromatic number of graph \(G\) as the minimum cost of a (proper) coloring of \(G\), and focus on the case that the marginal costs \(f(i+1)-f(i)\) are non-increasing. We provide bounds for general graphs, for specific classes of graphs, and for some operations on graphs. We also consider the number of colors used in an optimal coloring, and for example, characterize the trees where the bipartite coloring is not always optimal.