3-Coloring in Time O(1.3217^n)

TL;DR

This paper introduces a faster algorithm for 3-coloring graphs with a runtime of O(1.3217^n), improving upon previous methods by refining structural analysis to reduce computational complexity.

Contribution

It presents a novel approach that enhances the structural analysis used in 3-coloring algorithms, achieving a lower exponential runtime.

Findings

Achieved a 3-coloring algorithm with runtime O(1.3217^n)

Improved structural understanding of graph coloring problems

Reduced the exponential base in the algorithm's runtime

Abstract

We propose a new algorithm for 3-coloring that runs in time O(1.3217^n). For this algorithm, we make use of the time O(1.3289^n) algorithm for 3-coloring by Beigel and Eppstein. They described a structure in all graphs, whose vertices could be colored relatively easily. In this paper, we improve upon this structure and present new ways to determine how the involved vertices reduce the runtime of the algorithm.