Backbone colorings of graphs with bounded degree Original Research Article

We study backbone colorings, a variation on classical vertex colorings: Given a graph image and a subgraph image of image (the backbone of image), a backbone coloring for image and image is a proper vertex image-coloring of image in which the colors assigned to adjacent vertices in image differ by at least image. The minimal image for which such a coloring exists is called the backbone chromatic number of image. We show that for a graph image of maximum degree image where the backbone graph is a image-degenerated subgraph of image, the backbone chromatic number is at most image and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most image. We also present examples where these bounds are attained.