ManuscriptPreprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11Click May here 2018 to download doi:10.20944/preprints201805.0178.v1 Manuscript chromnum_soft_computing.docx Click here to view linked References 1Graph Coloring: A Novel Heuristic based on 1 2Trailing Path; Properties, Perspective and 2 3 3Applications in Structured Networks 4 5 4 6 5Abhirup Bandyopadhyay1$, Amit kumar Dhar2, Sankar Basu3$* 7 6 8 1 9 7 Department of Mathematics, National Institute of Technology, Durgapur 10 8Mahatma Gandhi Avenue, Durgapur 713209, West Bengal, India 11 9 2 Department of IT, IIIT Alahabad, Jhalwa, Alahabad 211012 12 10 3 Department of Physics and Astronomy, Clemson University, South Carolina, US 13 11 14 $ 15 12 These authors contributed equally to the work 16 13* Corresponding author 17 14 18 15C orrespondence: 19 20 16Email: [email protected]; Tel: +1-864-633-8394; ORCID: 0000-0003-1393-1982 21 17 22 23 18Abstract: Graph coloring is a manifestation of graph partitioning, wherein, a graph is 24 19partitioned based on the adjacency of its elements. Partitioning serves potentially as a 25 20compartmentalization for any structural problem. Vertex coloring is the heart of the 26 27 21probl em which is to find the chromatic number of a graph. The fact that there is no general 28 22efficient solution to this problem that may work unequivocally for all graphs opens up the 29 23realistic scope for combinatorial optimization algorithms to be invoked. The algorithmic 30 24complexity of graph coloring is non-deterministic in polynomial time (NP) and hard. To the 31 32 25best of our knowledge, there is no algorithm as yet that procures an exact solution of the 33 26chromatic number comprehensively for any and all graphs within the polynomial (P) time 34 27domain. However, several heuristics as well as some approximation algorithms have been 35 28attempted to obtain an approximate solution for the same. Here, we present a novel 36 29heuristic, namely, the 'trailing path', which returns an approximate solution of the 37 38 30chromatic number within polynomial time, and, with a better accuracy than most existing 39 31algorithms. The ‘trailing path’ algorithm is effectively a subtle combination of the search 40 32patterns of two existing heuristics (DSATUR and Largest First), and, operates along a 41 33trailing path of consecutively connected nodes (and thereby effectively maps to the problem 42 43 34of finding spanning tree(s) of the graph) during the entire course of coloring, where 44 35essentially lies both the novelty and the apt of the current approach. The study also 45 36suggests that the judicious implementation of randomness is one of the keys towards 46 37rendering an improved accuracy in such combinatorial optimization algorithms. Apart from 47 38the algorithmic attributes, essential properties of graph partitioning in random and 48 49 39different structured networks have also been surveyed, followed by a comparative study. 50 40The study reveals the remarkable stability and absorptive property of chromatic number 51 41across a wide array of graphs. Finally, a case study is presented to demonstrate the 52 42potential use of graph coloring in protein design – yet another hard problem in structural 53 43and evolutionary biology. The software is made available at: 54 55 44https://github.com/nemo8130/Chromnum 56 57 45Keywords: Chromatic Number, Graph Partitioning, NP to P, Trailing Path, Motif Identifier, 58 46Protein Design 59 60 61 62 63 1 64 65 © 2018 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 May 2018 doi:10.20944/preprints201805.0178.v1 471. Introduction 48In graph theory, graph coloring (Jensen and Toft 2011) is a special case of graph labeling 1 49(D íaz et al. 2002). It is an assignment of labels (Gallian 2015) traditionally known as 2 50assignment of 'colors' to edges and/or vertices of a graph subject to certain constraints. In 3 4 51trivial formalism, it is a way of coloring the vertices (nodes) of an undirected graph such 5 52that no two adjacent vertices could be assigned the same color. This is called vertex 6 53coloring (MacDougall et al. 2002). Similarly, an edge coloring (Wallis et al. 2000) assigns a 7 54color to each edge so that no two adjacent edges share the same color, and a face coloring 8 9 55of a planar graph (Sanders and Zhao) assigns colors to each face or region so that no two 10 56faces which share a common boundary share the same color. Given all this, vertex coloring 11 57remains the first chapter of the subject, and other coloring problems are transformable into 12 58a vertex version. For example, an edge coloring of a graph is actually a vertex coloring of 13 59the corresponding line graph, and a face coloring of a planer graph is a vertex coloring of 14 15 60its dual graph. However, non-vertex coloring problems are often stated and studied 16 61independently. That is partly because of perspective, and partly because some natural 17 62extension of problems which could be best studied in the form of edges. 18 63 19 64The convention of using colors was originated from coloring the countries of a 20 21 65geographical map, where each face is literally colored. This particular problem was 22 66formalized as coloring the faces of a planer graph. By implementing 'planar duality', a 23 67characteristic feature of planer graphs, the problem reduces to coloring of their vertices. As 24 68a generalization the face coloring problem could be viewed as vertex coloring problem of 25 26 69its dual graph. For the sake of simplicity and computational efficiency, first few positive or 27 70nonnegative integers are used as the "colors" (Zhang 2015) without the loss of generality, 28 71so that one can use any finite set of colors as the 'color set'. 29 72 30 73Graph coloring could be viewed as the problem of assignment of colors to a graph subject 31 32 74to number of constraints. Different constraints could range from constraints on a sub graph 33 75to those on the full graph or even on the color itself. The face coloring problem even 34 76attained popularity among common people in the form of the popular number puzzle 35 77Sudoku , the traveling salesman problem and the Chinese postman problem. One of the 36 37 78major applications of graph coloring is the register allocation in compilers. The range of 38 79applications grows even further ranging from coding theory to X-ray Crystallography 39 80(Blum et al. 1987), from radar and astronomy (Zarrazola et al. 2011) to circuit design and 40 81communication networks. Day-to-day real-life problems like guarding an art gallery, 41 82physical layout segmentation, round robin sports scheduling, aircraft scheduling (Marx 42 43 832003) etc. should potentially be benefited by an elegant algorithmic solution of the 44 84problem. Graph coloring is still a very active area of research with a bunch of unsolved 45 85problems, e.g., the chromatic number of the plane is unknown where two points are 46 86adjacent if they have unit distance. Other open problems concerning the chromatic number 47 87of graphs include the Hadwiger's conjecture (Bollobás et al. 1980) stating that every graph 48 49 88with chromatic number k has a k-complete sub graph with k vertices; the Erdős–Faber– 50 89Lovász conjecture bounding the chromatic number of unions of complete graphs that have 51 90exactly one vertex in common to each pair, and the Albertson conjecture (Albertson et al. 52 912010) that among k-chromatic graphs the complete graphs are the ones with the smallest 53 54 92crossing number. 55 93 56 94The first results about graph coloring deal almost exclusively with planar graphs in the 57 95form of the coloring of maps (Stiebitz and Škrekovski 2006). While working on the map 58 96coloring problem of the counties of England, Francis Guthrie postulated the four color 59 60 97conjecture, noting that four colors were sufficient to color a map so that no regions sharing 61 62 63 2 64 65 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 May 2018 doi:10.20944/preprints201805.0178.v1 98a common border receives the same color. In 1879, Alfred Kempe published a paper 99(Kempe 1879) that claimed to establish the result which was controversial and much debate 100followed. In fact it took close to a century until the four color theorem was finally proved in 1 2 1011976 by Kenneth Appel and Wolfgang Haken (Appel and Haken 1977). The proof was the 3 102first major computer-aided proof in this problem which went back to the ideas of Heawood 4 103and Kempe while largely disregarding the intervening developments. From that time 5 104onwards, active research is ongoing on the algorithmic attributes of graph coloring. The 6 7 105chromatic number problem falls in the list of Karp's 21 NP-complete problems (Karp 8 1061972), and remains computationally NP-hard (Garey et al. 1974). That is to say that, it is 9 107NP -complete to decide whether a given graph admits a k-coloring for any given k except for 10 108the trivial cases k ∈ {0,1,2}. In other words, the 3-coloring problem remains NP-complete 11 109even on 4-regular planer graphs (Dailey 1980), and the approximation algorithm 12 13 110(Hallórsson 1993), the most established one in the field, computes a coloring of graph-size 14 111n at most within a factor of O(n(log n)−3(log (log (n)))2) of the chromatic number. 15 112 16 17 113The relatively recent concept of chromatic polynomial (Dong et al. 2005) has provided 18 114another alternative approach towards solving the graph coloring problem, serving important 19 115fundamental structures in algebraic graph theory.