What is the most common type of simple graph Colouring problem?

A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex. In other words a simple graph is a graph without loops and multiple edges. Two vertices are said to be adjacent if there is an edge (arc) connecting them.

How do you prove edge coloring?

In any proper edge-colouring, the d(v) edges that are incident with v, must all be assigned different colours. Thus, any proper edge-colouring must have at least d(v)=∆(G) distinct colours. This means χ′(G)≥∆(G).

What is the main idea of graph coloring problem explain with example?

Graph coloring is the procedure of assignment of colors to each vertex of a graph G such that no adjacent vertices get same color. The objective is to minimize the number of colors while coloring a graph. The smallest number of colors required to color a graph G is called its chromatic number of that graph.

How many edges does a simple graph have?

A graph with no loops and no parallel edges is called a simple graph. The maximum number of edges possible in a single graph with ‘n’ vertices is nC2 where nC2 = n(n – 1)/2. The number of simple graphs possible with ‘n’ vertices = 2nc2 = 2n(n-1)/2.

What is graph coloring explain with example?

What is equitable coloring in graph theory?

In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that No two adjacent vertices have the same color, and The numbers of vertices in any two color classes differ by at most one.

What is the smallest number of colors in an equitable coloring?

The smallest number of colors in an equitable coloring of this graph is four, as shown in the illustration: the central vertex must be the only vertex in its color class, so the other five vertices must be split among at least three color classes in order to ensure that the other color classes all have at most two vertices.

What is the difference between equitable chromatic number and equitable threshold?

The equitable chromatic number of a graph G is the smallest number k such that G has an equitable coloring with k colors. But G might not have equitable colorings for some larger numbers of colors; the equitable chromatic threshold of G is the smallest k such that G has equitable colorings for any number of colors greater than or equal to k.