On critical trees labeled with a condition at distance two Original Research Article

An image-labeling of a graph is an assignment of nonnegative integers to its vertices so that adjacent vertices get labels at least two apart and vertices at distance two get distinct labels. A graph is said to be image-critical if image is the minimum span taken over all of its image-labelings, and every proper subgraph has an image-labeling with span strictly smaller than image. Georges and Mauro have studied 5-critical trees with maximum degree image by examining their path-like substructures. They also presented an infinite family of 5-critical trees of maximum degree image. We generalize these results for image-critical trees with image.