Efficiently coloring an arbitrary graph is a fundamental and notoriously difficult algorithmic problem. This talk focuses on the restricted problem of determining the complexity of coloring graphs which do not contain a certain induced subgraph. Combining results of Kamiński and Lozin, and Hoyler, it follows that this problem remains NP-complete unless the excluded induced subgraph is a disjoin...

The graph coloring problem is a notoriously hard problem, for which we do not have efficient algorithms. A coloring of a graph is an assignment of colors to its vertices such that the end points of every edge have different colors. A k-coloring is a coloring that uses at most k distinct colors. The graph coloring problem is to find a coloring that uses the minimum number of colors. Given a 3-co...

This work proposes an efficient method for solving the Distance-1 Edge Coloring problem (D1EC) for the assignment of orthogonal channels in wireless networks with changing topology. The coloring algorithm is performed by means of the simulated annealing method, a generalization of Monte Carlo methods for solving combinatorial problems. We show that the simulated annealing-based coloring converg...

A graph edge is d-coloring redundant if the removal of the edge does not change the set of dcolorings of the graph. Graphs that are too sparse or too dense do not have coloring redundant edges. Tight upper and lower bounds on the number of edges in a graph in order for the graph to have a coloring redundant edge are proven. Two constructions link the class of graphs with a coloring redundant ed...

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 ...