Improper C-colorings of graphs Original Research Article

For an integer image, the image-improper upper chromatic number image of a graph image is introduced here as the maximum number of colors permitted to color the vertices of image such that, for any vertex image in image, at most image vertices in the neighborhood image of image receive colors different from that of image. The exact value of image 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 image 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 image is studied, and structural correspondence between image-improper C-colorings and certain kinds of edge cuts is shown.