A Brooks-type bound for squares of K4K4-minor-free graphs

Refining a bound by Lih, Wang and Zhu, we prove that if the square G2G2 of a K4K4-minor-free graph GG with maximum degree Δ⩾6Δ⩾6 does not contain a complete subgraph on ⌊32Δ⌋+1 vertices, then G2G2 is ⌊32Δ⌋-colorable.