Labeling planar graphs with a condition at distance two
Peter Bella, Daniel Král͏̌, Bojan Mohar, Katarina Quittnerová · Discrete Mathematics & Theoretical Computer Science · 2005
An $L(2,1)$-labeling of a graph is a mapping $c:V(G) \to \{0,\ldots,K\}$ such that the labels assigned to neighboring vertices differ by at least $2$ and the labels of vertices at distance two are different. Griggs and Yeh [SIAM J. Discrete Math. 5 (1992), 586―595] conjectured that every graph $G$ with maximum degree $\Delta$ has an $L(2,1)$-labeling with $K \leq \Delta^2$. We verify the conjecture for planar graphs with maximum degree $\Delta eq 3$.