Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Graph Coloring: A Novel Heuristic Based on Trailing Path; Properties, Perspective and Applications in Structured Networks

Version 1 : Received: 10 May 2018 / Approved: 11 May 2018 / Online: 11 May 2018 (08:58:35 CEST)

A peer-reviewed article of this Preprint also exists.

Bandyopadhyay, A., Dhar, A.K. & Basu, S. Soft Comput. (2019). https://doi.org/10.1007/s00500-019-04278-8 Bandyopadhyay, A., Dhar, A.K. & Basu, S. Soft Comput. (2019). https://doi.org/10.1007/s00500-019-04278-8

## Abstract

Graph coloring is a manifestation of graph partitioning, wherein, a graph is partitioned based on the adjacency of its elements. Partitioning serves potentially as a compartmentalization for any structural problem. Vertex coloring is the heart of the problem which is to find the chromatic number of a graph. The fact that there is no general efficient solution to this problem that may work unequivocally for all graphs opens up the realistic scope for combinatorial optimization algorithms to be invoked. The algorithmic complexity of graph coloring is non-deterministic in polynomial time (NP) and hard. To the best of our knowledge, there is no algorithm as yet that procures an exact solution of the chromatic number comprehensively for any and all graphs within the polynomial (P) time domain. However, several heuristics as well as some approximation algorithms have been attempted to obtain an approximate solution for the same. Here, we present a novel heuristic, namely, the 'trailing path', which returns an approximate solution of the chromatic number within polynomial time, and, with a better accuracy than most existing algorithms. The ‘trailing path’ algorithm is effectively a subtle combination of the search patterns of two existing heuristics (DSATUR and Largest First), and, operates along a trailing path of consecutively connected nodes (and thereby effectively maps to the problem of finding spanning tree(s) of the graph) during the entire course of coloring, where essentially lies both the novelty and the apt of the current approach. The study also suggests that the judicious implementation of randomness is one of the keys towards rendering an improved accuracy in such combinatorial optimization algorithms. Apart from the algorithmic attributes, essential properties of graph partitioning in random and different structured networks have also been surveyed, followed by a comparative study. The study reveals the remarkable stability and absorptive property of chromatic number across a wide array of graphs. Finally, a case study is presented to demonstrate the potential use of graph coloring in protein design – yet another hard problem in structural and evolutionary biology.

## Keywords

chromatic number; graph partitioning; NP to P; motif identifier; protein design

## Subject

Computer Science and Mathematics, Computer Vision and Graphics

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