Generalized H-Coloring of Graphs

Abstract

For fixed simple graph H and subsets of natural numbers σ and ρ, we introduce (H, σ, ρ)-colorings as generalizations of H-colorings of graphs. An (H, σ, ρ)-coloring of a graph G can be seen as a mapping f : V (G) → V (H), such that the neighbors of any v ∈ V (G) are mapped to the closed neighborhood of f(v), with σ constraining the number of neighbors mapped to f(v), and ρ constraining the number of neighbors mapped to each neighbor of f(v). A traditional H-coloring is in this sense an (H, {0}, {0, 1, ...})-coloring. We initiate the study of how these colorings are related and then focus on the problem of deciding if an input graph G has an (H; {0}, {1, 2, ...})-coloring. This H-COLORDOMINATION problem is shown to be no easier than the H-COVER problem and NP-complete for various infinite classes of graphs.