What is proper coloring in graph theory?

A proper coloring is an as- signment of colors to the vertices of a graph so that no two adjacent vertices have the same 1 Page 2 color. A k-coloring of a graph is a proper coloring involving a total of k colors. A graph that has a k-coloring is said to be k-colorable.

What is the condition in coloring of a graph?

Abstract. For an integer r > 0 , a conditional ( k , r ) -coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex of degree at least r in G will be adjacent to vertices with at least r different colors.

How do you color a graph?

Method to Color a Graph

1. Step 1 − Arrange the vertices of the graph in some order.
2. Step 2 − Choose the first vertex and color it with the first color.
3. Step 3 − Choose the next vertex and color it with the lowest numbered color that has not been colored on any vertices adjacent to it.
4. Example.

What do you need to know about coloring a graph?

Graph coloring is nothing but a simple way of labelling graph components such as vertices, edges, and regions under some constraints. In a graph, no two adjacent vertices, adjacent edges, or adjacent regions are colored with minimum number of colors.

How to tell if a graph is a tree?

However, there’s another simple method which we can use to see whether the given graph is a tree or not. All trees have N – 1 edges, where N is the number of nodes.

Is the four color theorem equivalent to Tait coloring?

The four color theorem is equivalent to the assertion that every planar cubic bridgeless graph admits a Tait coloring. Total coloring is a type of coloring on the vertices and edges of a graph.

What is the chromatic number of vertex coloring in graph theory?

Simply put, no two vertices of an edge should be of the same color. The minimum number of colors required for vertex coloring of graph ‘G’ is called as the chromatic number of G, denoted by X (G). χ (G) = 1 if and only if ’G’ is a null graph. If ’G’ is not a null graph, then χ (G) ≥ 2