An edge-coloring of a graph G1⁄4 ðV ; EÞ is a function c that assigns an integer c(e) (called color) in f 0;1;2;...g to every edge eAE so that adjacent edges are assigned different colors. An edge-coloring is compact if the colors of the edges incident to every vertex form a set of consecutive integers. The deficiency problem is to determine the minimum number of pendant edges that must be adde...

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...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. Howeve...

A coloring of a graph G is an assignment of colors to the vertices of G such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring of G is a coloring such that no cycle of G receives exactly two colors, and the acyclic chromatic number χA(G) of a graph G is the minimum number of colors in any such coloring of G. Given a graph G and an integer k, determini...

A k-edge-coloring of a graph G = (V, E) is a function c that assigns an integer c(e) (called color) in {0, 1, · · · , k−1} to every edge e ∈ E so that adjacent edges get different colors. A k-edge-coloring is linear compact if the colors incident to every vertex are consecutive. The problem k − LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring...

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...

The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of...

modeling of network security is useful approach to comprehend the status. In this paper, network is modeled in a graph. Security problem is solved in graph as Graph Coloring Problem (GCP). In GCP, two adjacent nodes must have different colors. Thus GCP provides the security in the network. One objective in GCP is chromatic number and another objective is total price. We present a multi-objectiv...

We revisit in this paper the stochastic model for minimum graph-coloring introduced in (C. Murat and V. Th. Paschos, On the probabilistic minimum coloring and minimum k-coloring, Discrete Applied Mathematics 154, 2006), and study the underlying combinatorial optimization problem (called probabilistic coloring) in bipartite and split graphs. We show that the obvious 2-coloring of any connected b...

Graph coloring problem is a well-known NP-complete problem and there are many approaches proposed to solve this problem. For a graph coloring algorithm to be efficient, it must color the input graph with minimum colors and must also find the solution in the minimum possible time. Heuristic approaches emphasize on the time complexity while the exact approaches concentrate on the number of colors...