Griggs–Yeh L(2,1)-labeling conjecture

Let $G$ be a graph with maximum degree $\Delta$, and let $\lambda^{2,1}(G)$ be the minimum span of an $L(2,1)$-labeling, in which vertices at distance one receive labels differing by at least $2$ and vertices at distance two receive distinct labels. Griggs–Yeh conjecture.

$$\lambda^{2,1}(G)\le\Delta^2.$$

This conjecture is the labeling analogue of quadratic coloring bounds for graph squares; the survey does not state a general resolution.