Labeling graphs with two distance constraints Original Research Article

Given a graph image and integers image and image, with image, image, an image-labeling of image is a function image such that image if image and image if image. A image-labeling is an image-labeling image such that image. The image-labeling number of image, denoted by image, is the smallest number image such that image has a image-image-labeling. In this paper, we give upper bounds and lower bounds of the image-labeling number for general graphs and some special graphs. We also discuss the image-labeling number of image, when image is a path, a power of a path, or Cartesian product of two paths.