K Edge Coloring In Graph Theory, Graph Theory

Graph edge coloring has a rich theory many applications and beautiful conjectures and it is studied not only by mathematicians but also by computer scientists.

Natural blue food coloring. A k edge coloring of g is an assignment of k colors to the edges of g in such a way that any two edges meeting at a common vertex are assigned different colors. Consider this example with k 4. Two edges are said to be adjacent if they are connected to the same vertex.

In graph theory edge coloring of a graph is an assignment of colors to the edges of the graph so that no two adjacent edges have the same color with an optimal number of colors. Let g be a graph with no loops. Soifer 2008 provides the following geometric construction of a coloring in this case.

A complete graph k n with n vertices is edge colorable with n 1 colors when n is an even number. In its simplest form it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color. Graph coloring is one of the most important concepts in graph theory.

K n 1. There is no known polynomial time algorithm for edge coloring every graph with an. Hence the chromatic number of k n n.

If g is the complete graph k n then p g k kk 1k 2. Besides known results a new basic result about brooms is obtained. In this survey written for the non expert we shall describe some main results and techniques and state some of the many popular conjectures in the theory.

If g has a k edge coloring then g is said to be k edge. Place n points at the vertices and center of a regular n 1 sided polygonfor each color class include one edge from the center to one of the polygon. This is called a vertex coloringsimilarly an edge coloring assigns a color to each.

Applications of graph coloring. It is an assignment of labels traditionally called colors to elements of a graph subject to certain constraints. In the complete graph each vertex is adjacent to remaining n 1 vertices.

In graph theory graph coloring is a special case of graph labeling. This is a special case of baranyais theorem. 1 a bipartite graph g has a k edge coloring in which all k colors appear at each vertex.

Hence each vertex requires a new color. Gupta proved the two following interesting results.