On the square coloring of comparability graphs

We find sufficient conditions for the square of a comparability graph Comp(P) of a poset P to be (Delta + r)-colorable when Comp(P) lacks K-2(,r) for some r >= 1. Furthermore, we show that the problem of coloring the square of the comparability graph of a poset of height at least four can be reduced to the case of height three, where the height of a poset is the size of a maximum chain.