On critical trees labeled with a condition at distance two

Abstract: An -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 -critical if is the minimum span taken over all of its -labelings, and every proper subgraph has an -labeling with span strictly smaller than . Georges and Mauro have studied 5-critical trees with maximum degree by examining their path-like substructures. They also presented an infinite family of 5-critical trees of maximum degree . We generalize these results for -critical trees with . [Copyright &y& Elsevier]