Chromatic Sums for Colorings Avoiding Monochromatic Subgraphs

Given graphs G and H, a vertex coloring c:V(G)→N is an H-free coloring of G if no color class contains a subgraph isomorphic to H. The H-free chromatic number of G, χ(H,G), is the minimum number of colors in an H-free coloring of G. The H-free chromatic sum of G,Σ(H,G), is the minimum value achieved by summing the vertex colors of each H-free coloring of G. We provide a general bound for Σ(H,G), discuss the computational complexity of finding this parameter for different choices of H, and prove an exact formulas for some graphs G. For every integer k and for every graph H, we construct families of graphs, Gk with the property that k more colors than χ(H,G) are required to realize Σ(H,G) for H-free colorings. More complexity results and constructions of graphs requiring extra colors are given for planar and outerplanar graphs.