On critical trees labeled with a condition at distance two

An L(2,1)-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 λ-critical if λ is the minimum span taken over all of its L(2,1)-labelings, and every proper subgraph has an L(2,1)-labeling with span strictly smaller than λ. Georges and Mauro have studied 5-critical trees with maximum degree Δ=3 by examining their path-like substructures. They also presented an infinite family of 5-critical trees of maximum degree Δ=3. We generalize these results for λ-critical trees with Δ⩾4.