This paper is concerned with a refutation system (proof system) for a class of linear constraint systems called difference constraint systems (DCS). In particular, we study the refutability of DCSs in the unit refutation (UR) system. Recall that a difference constraint is a linear relationship of the form: x_i-x_j ≤ b_ij and a DCS is a conjunction of such constraints. Associated with a refutation system are three important features, viz., (a) Soundness, (b) Completeness, and (c) Efficiency. The UR system is sound and efficient; however, it is incomplete, in that unsatisfiable DCSs may not have unit refutations. We establish that this refutation system is efficient in that there exists a tractable algorithm for determining if a DCS has a UR. Investigating weak (incomplete) refutation systems leads to a better understanding of the inference rules required for establishing contradictions in the given constraint system. Thus, this study is well-motivated. Despite the fact that unit refutations can be exponentially long in terms of the input system size, we provide a compact representation of these refutations. This compact representation is an important contribution of this paper.