A Polynomial-Time Algorithm for Coloring Perfect Graphs Based on Walk Counting
Data Structures and Algorithms
2026-07-10 v1 Discrete Mathematics
Combinatorics
Optimization and Control
Abstract
We present a polynomial-time algorithm for optimally coloring perfect graphs that is based entirely on graph-theoretic operations. At its core, the algorithm decides whether a perfect graph contains a clique of a given size by iteratively counting walks in the graph with certain weights assigned to its edges and nonedges. These weights are initialized according to a uniform scheme and then updated in each iteration based on the walk counts from the previous iteration.
Cite
@article{arxiv.2607.09309,
title = {A Polynomial-Time Algorithm for Coloring Perfect Graphs Based on Walk Counting},
author = {Amir Ali Ahmadi and Pravesh K. Kothari and Yukai Tang},
journal= {arXiv preprint arXiv:2607.09309},
year = {2026}
}