A bound on the chromatic number of the square of a planar graph

Abstract

Wegner conjectured that the chromatic number of the square of any planar graph G with maximum degree $\mathrm{\Delta }⩾8$ is bounded by $\chi (G^2)⩽\lfloor \frac{3}{2}\mathrm{\Delta }\rfloor +1$. We prove the bound $\chi (G^2)⩽\lceil \frac{5}{3}\mathrm{\Delta }\rceil +78$. This is asymptotically an improvement on the previously best-known bound. For large values of $\mathrm{\Delta }$ we give the bound of $\chi (G^2)⩽\lceil \frac{5}{3}\mathrm{\Delta }\rceil +25$. We generalize this result to $L(p,q)$-labeling of planar graphs, by showing that ${\lambda }_q^p(G)⩽q\lceil \frac{5}{3}\mathrm{\Delta }\rceil +18p+77q-18$. For each of the results, the proof provides a quadratic time algorithm.