Asia Mathematika Volume: 2 Issue: 2 , (2018) Pages: 33 – 40 Available online at www.asiamath.org Computation of First-Fit Coloring of Graphs V. Yegnanarayanan * School of Humanities and Sciences, SASTRA University, Thanjavur-613401, TN, India Received: 10 Oct 2018 • Accepted: 9 Jan 2019 • Published Online: 5 Apr 2019 Abstract: The first-fit chromatic number of a graph is the maximum possible number of colors used in a first fit coloring of the graph. In this paper, we compute the first-fit chromatic number for a special class of bipartite graphs. Further we give a crisp description on the computational aspects of the first fit chromatic number and indicate the scope for further applications. We also raise some open problems. Key words: Graph, coloring, chromatic number, First-Fit Chromatic Number 1. Introduction A coloring or a proper coloring of a graph G is an assignment of positive integers called “colors” to the vertices of G so that adjacent vertices have different colors. The minimum number of colors used in a proper coloring is called the chromatic number of G , denoted by χ(G). A graph coloring can be a solution to some specific kind of problem. For instance, an animal preserving firm is destined to conserve a set V of animals with different compatibility levels among themselves. We distinguish an incompatible pair with an edge. A vertex coloring of the graph G(V,E) allots enclosures to them. Coloring of such a graph with many colors are easy to obtain but we require a coloring with least number of colors. The name coloring hails from the 19th century challenge that whether four colors are sufficient to color a planar map of a country. Each state is a vertex, and every pair of states that share a common boundary becomes an edge, so a proper coloring gives them different colors. The identities of the colors are not pertinent mathematically, so what is desired is not a coloring, and is actually a graph partition that honors graph structure. Graph coloring is one of the most vital concepts in graph theory. It is employed in several practical applications of computer science such as clustering, data mining, image capturing, image segmentation, networking, resource allocation, processes scheduling etc., Several upper bounds on the chromatic number comes from algorithms that generates a coloring. The most basic algorithm is the greedy algorithm. A greedy coloring relative to a vertex ordering σ = v 1 <v 2 < ··· <v n of V (G) is obtained by coloring the vertices in the order v 1 ,v 2 ,...,v n assigning to v i the smallest positive integer not already used on its neighbours which are lower-indexed. The first-fit chromatic number, denoted by χ FF (G), is the largest number of colors used such that G has a greedy coloring. The study of first-fit chromatic number continues to attract the attention of researchers since a first- fit coloring problem occurs frequently in real life applications such as dynamic storage allocation problem [4, 7, 9, 10, 12, 16] and radio frequency allocation problem [19]. Historically, the first-fit chromatic number is also called the Grundy number. The study of Grundy coloring dates back to the 1930’s when Grundy used them in the study of kernels of directed Graphs [11]. After that, many researchers have studied the first-fit chromatic number under different names [2, 6]. It is believed that Christen and Selkow were the first to define c Asia Mathematika * Correspondence: prof.yegna@gmail.com 33