Improper C-colorings of graphs

For an integer k≥1, the k-improper upper chromatic numberχ̄k-imp(G) of a graph G is introduced here as the maximum number of colors permitted to color the vertices of G such that, for any vertex v in G, at most k vertices in the neighborhood N(v) of v receive colors different from that of v. The exact value of χ̄k-imp is determined for several types of graphs, and general estimates are given in terms of various graph invariants, e.g. minimum and maximum degree, vertex covering number, domination number and neighborhood number. Along with bounds on χ̄k-imp for Cartesian products of graphs, exact results are found for hypercubes. Also, the analogue of the Nordhaus-Gaddum theorem is proved. Moreover, the algorithmic complexity of determining χ̄k-imp is studied, and structural correspondence between k-improper C-colorings and certain kinds of edge cuts is shown.