Coloring the square of a K4-minor free graph

Let G be a K4-minor free graph with maximum degree Δ. We prove that the chromatic number of the square of G is at most (i) Δ+3 if 2⩽Δ⩽3; or (ii) ⌊3Δ/2⌋+1 if Δ⩾4. Examples are given to show the bounds can be attained.