# SAT Encoding of Partial Ordering Models for Graph Coloring Problems

arXiv: https://arxiv.org/abs/2403.15961 · HTML: https://mlsanity.com/paper/2403.15961/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 24 references that are in the map.
- First posted on arXiv 23 Mar 2024.

## Builds on (most cited first, 1 of 1)

- [New Integer Linear Programming Models for the Vertex Coloring Problem](https://mlsanity.com/paper/1706.10191/) (2017)

## Built on by (most cited first, 1 of 1)

- [A Customized SAT-based Solver for Graph Coloring](https://mlsanity.com/paper/2504.04821/) (2025)

## Abstract (arXiv)

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.