American Journal of Mathematics and Statistics 2 (4): 439-441, 2006 ISSN 1549-3636 © 2006 Science Publications Corresponding Author: Dr. Hussein Al-Omari, 1 Computer Science Department, School of Computer Science and Information Technology, Applied Science University, Amman, Jordan 439 New Graph Coloring Algorithms Dr. Hussein Al-Omari and Khair Eddin Sabri 1 Computer Science Department, School of Computer Science and Information Technology Applied Science University, Amman, Jordan 2 Computer Science Department, King Abdullah II School of Information Technology University of Jordan, Amman, Jordan Abstract: Two new heuristic graph-coloring algorithms, based on known heuristic algorithms, have been introduced. One of them is a modification of the Largest Degree Ordering (LDO) algorithm, and the other one is a modification of the Saturation Degree Ordering (SDO) algorithm. The two new algorithms proposed in this paper, were compared empirically, in terms of used colors, with some of the known heuristic graph-coloring algorithms such as: Largest Degree Ordering (LDO), First Fit (FF), Saturated Degree Ordering (SDO), and Incident Degree Ordering (IDO). As a result of this comparison, it was found that the proposed algorithms were better than the original ones with respect to the number of used colors. Key words: Graph coloring algorithms, degree ordering, first-fit algorithm INTRODUCTION Graph coloring is defined as coloring the nodes of a graph with the minimum number of colors without any two adjacent nodes having the same color. For example, the linked list needs two colors and so does the binary search tree. Graph coloring has wide applications such as: estimation of sparse Jacobins, scheduling and registering allocation. The coloring of a graph G = (V, E) is a mapping c: v ﾆ s, where “s” is a finite set of colors, such that if vw ∈ E then c(v) ≠ c(w). In other words, adjacent vertices are not assigned the same color [1] . The problem that arises is the coloring of a graph provided that no adjacent vertices have the same color. The chromatic number X(G) is the minimum number of colors needed for a coloring of G. A graph G is k_chromatic, if X(G) = k, and G is k_colorable, if X(G) ≤ k.. Graph coloring is one of the most useful models in graph theory. It has been used to solve problems in school timetabling, computer register allocation, electronic bandwidth allocation, and many other applications [2] . In this paper, two new heuristic graph-coloring algorithms, based on known heuristic algorithms, have been introduced. One of them is a modification of the Largest Degree Ordering (LDO) algorithm, and the other one is a modification of the Saturation Degree Ordering (SDO) algorithm. The two new algorithms proposed in this paper, were compared empirically, in terms of used colors, with some of the known heuristic graph-coloring algorithms such as: Largest Degree Ordering (LDO), First Fit (FF), Saturated Degree Ordering (SDO), and Incident Degree Ordering (IDO). As a result of this comparison, it was found that the proposed algorithms were better than the original ones with respect to the number of used colors. Here, we investigated some of the heuristic graph coloring algorithms (FF, FDO, SDO, and LDO), and then by modifying some of them, we introduced two new modified algorithms. Graph coloring algorithms: There are many heuristic sequential techniques for coloring a graph. One of them is the Greedy Graph Coloring. This technique focuses on carefully picking the next vertex to be colored. In this heuristic algorithm, once a vertex is colored, its color never changes. Below, we explain the first fit and degree based ordering techniques. a. First fit: First Fit algorithm is the easiest and fastest technique of all greedy coloring heuristics. The algorithm sequentially assigns each vertex the lowest legal color. This algorithm has the advantage of being very simple and fast and can be implemented to run in O(n) [2,3] . b. Degree based ordering: It provides a better strategy for coloring a graph. It uses a certain selection criterion for choosing the vertex to be colored. This strategy is better than the First Fit which simply picks a vertex from an arbitrary order. Some strategies for selecting the next vertex to be colored have been proposed such as: Largest degree ordering (LDO): It chooses a vertex with the highest number of neighbors. Intuitively, LDO