Labeling the image-path with a condition at distance two Original Research Article

For integer image, the infinite image-path image is the graph on vertices image such that image is adjacent to image if and only if image. The image-path on image vertices is the subgraph of image induced by vertices image. For non-negative reals image and image, a image-labeling of a simple graph image is an assignment of non-negative reals to the vertices of image such that adjacent vertices receive reals that differ by at least image, vertices at distance two receive reals that differ by at least image, and the absolute difference between the largest and smallest assigned reals is minimized. With image denoting that minimum difference, we derive image for image, image, and image. For image, we obtain upper bounds on image and use them to give image for image and image. We also determine image and image for all image.