Problem
Research questions and friction points this paper is trying to address.
perfect graphs
graph coloring
polynomial-time algorithm
clique detection
walk counting
A Polynomial-Time Algorithm for Coloring Perfect Graphs Based on Walk Counting
🤖 AI Summary
This work addresses the problem of optimal coloring of perfect graphs and presents the first polynomial-time algorithm based entirely on graph-theoretic operations. The approach initializes weighted edges and non-edges, then iteratively computes the number of weighted paths in the graph to detect the existence of cliques of a given size, progressively constructing an optimal coloring. Its key innovation lies in the seamless integration of clique detection with path-counting mechanisms, thereby avoiding reliance on traditional algebraic or optimization tools. This yields an efficient, purely combinatorial method for optimally coloring perfect graphs, offering significant theoretical and algorithmic contributions.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
perfect graphs
polynomial-time algorithm
graph coloring
walk counting
clique detection
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