A Brooks-type bound for squares of K4-minor-free graphs

Refining a bound by Lih, Wang and Zhu, we prove that if the square G^2 of a K"4-minor-free graph G with maximum degree @D>=6 does not contain a complete subgraph on @?32@D@?+1 vertices, then G^2 is @?32@D@?-colorable.