  In this paper, we suggest new SAT encodings of the partial-ordering based ILP
model for the graph coloring problem (GCP) and the bandwidth coloring problem
(BCP). The GCP asks for the minimum number of colors that can be assigned to
the vertices of a given graph such that each two adjacent vertices get
different colors. The BCP is a generalization, where each edge has a weight
that enforces a minimal "distance" between the assigned colors, and the goal is
to minimize the "largest" color used. For the widely studied GCP, we
experimentally compare our new SAT encoding to the state-of-the-art approaches
on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is
effective for sparse graphs and even outperforms the state-of-the-art on some
DIMACS instances. For the BCP, our theoretical analysis shows that the
partial-ordering based SAT and ILP formulations have an asymptotically smaller
size than that of the classical assignment-based model. Our practical
evaluation confirms not only a dominance compared to the assignment-based
encodings but also to the state-of-the-art approaches on a set of benchmark
instances. Up to our knowledge, we have solved several open instances of the
BCP from the literature for the first time.
