The (∆+1)-coloring problem is a fundamental symmetry breaking problem in distributed computing. We give a new randomized coloring algorithm for (∆ + 1)-coloring running in O( √ log ∆) + 2O( √ log logn) rounds with probability 1 − 1/nΩ(1) in a graph with n nodes and maximum degree ∆. This implies that the (∆ + 1)-coloring problem is easier than the maximal independent set problem and the maximal...

Graph coloring is the critical and the complex algorithmic problem used with different configuration and constraints for different applications. This paper has explored the graph coloring method in different aspects. Some of the new measures, methods and constraints are also defined for adaptive graph coloring. The theorem specific graph coloring is here defined to provide the constraint genera...

For a bounded integer , we wish to color all edges of a graph G so that any two edges within distance have different colors. Such a coloring is called a distance-edge-coloring or an -edge-coloring of G. The distance-edge-coloring problem is to compute the minimum number of colors required for a distance-edge-coloring of a given graph G. A partial k-tree is a graph with tree-width bounded by a f...

A twin edge k-coloring of a graph G is a proper edge coloring of G with the elements of Zk so that the induced vertex coloring in which the color of a vertex v in G is the sum (in Zk) of the colors of the edges incident with v is a proper vertex coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Among the results presented are formulas for th...

Graph coloring in general is an extremely easy-to-understand yet powerful tool. It has wide-ranging applications from register allocation to image segmentation. For such a simple problem, however, the question is surprisingly intractable. In this section I will introduce the problem formally, as well as present some general background on graph coloring. There are several ways to color a graph, ...

In this paper we investigate a variation of the graph coloring game, as studied in [2]. In the original coloring game, two players, Alice and Bob, alternate coloring vertices on a graph with legal colors from a fixed color set, where a color α is legal for a vertex if said vertex has no neighbors colored α. Other variations of the game change this definition of a legal color. For a fixed color ...

We prove that if is a subgraph of complete multipartite graph , then contains connected component satisfying . use this to every 3-coloring the edges monochromatic with at least edges. further show such coloring has circuit fraction This verifies conjecture Conlon and Tyomkyn. Moreover, for general we -coloring

In this paper, we study the coloring problem for the undirected binary de Bruijn interconnection network. The coloring scheme is simple and fast. We propose the coloring algorithm by using the pseudo shortestpath spanning tree rooted at (0 00). Each processor can find its color number by its own identity. Then, based on our coloring algorithm, we propose a 1-fair alternator. Our design is optim...

The circular arc coloring problem is to find a minimum coloring of a set of arcs of a circle so that no two overlapping arcs share a color. This N P-hard problem arises in a rich variety of applications and has been studied extensively. In this paper we present an O(n2m) combinatorial algorithm for optimally coloring any set of arcs that corresponds to a perfect graph, and propose a new approac...