COLORING THE SQUARE OF AN OUTERPLANAR GRAPH

Abstract

Let $G$ be an outerplanar graph with maximum degree $\Delta(G) \ge 3$. We prove that the chromatic number $\chi(G^2)$ of the square of $G$ is at most $\Delta(G)+2$. This confirms a conjecture of Wegner [8] for outerplanar graphs. The upper bound can be further reduced to the optimal value $\Delta(G)+1$ when $\Delta(G) \ge 7$.