# Recycling Solutions for Vertex Coloring Heuristics

## TL;DR

This paper introduces the recycle method, a simple initial solution generator for vertex coloring heuristics, which leverages (k+1)-colorings to improve the speed of finding legal k-colorings, supported by experimental evidence.

## Contribution

The paper proposes the recycle method as a novel initial solution generator that enhances heuristic algorithms for vertex coloring by using existing (k+1)-colorings.

## Key findings

- Recycle method improves heuristic speed in vertex coloring.
- Experimental results show faster convergence with the recycle method.
- Recommended as standard initial solution generator for various algorithms.

## Abstract

The vertex coloring problem is a well-known NP-hard problem and has many applications in operations research and in scheduling. A conventional approach to the problem solves the k-colorability problem iteratively, decreasing k one by one. Whether a heuristic algorithm finds a legal k-coloring quickly or not is largely affected by an initial solution. We highlight a simple initial solution generator, which we call the recycle method, which makes use of the legal (k+1)-coloring that has been found. An initial solution generated by the method is expected to guide a heuristic algorithm to find a legal k-coloring more quickly than conventional methods, as demonstrated by experimental studies. The results suggest that the recycle method should be used as the standard initial solution generator for both local search algorithms and modern hybrid methods.