Question

Show that if G is a graph with n vertices, then no more than n/2 edges can be colored the same in an edge coloring of G.

Short answer

Here n is the numbers of vertices, so we have n/2 edges colored the same

Step-by-step solution

Given Information.

G is a graph with n vertices, then no more than n/2 edges can be colored the same in an edge coloring of G.

Definition of Edge chromatic numbers and edge coloring.

The edge chromatic numbers, sometimes also called the chromatic index, of a

graph is the fewest number of colors necessary to color each edge of. such that

no two edges incident on the same vertex have the same color.

Edge coloring means assigning different colors to the edges of the same vertex.

Explanation.

Since we have the graph with n vertices, and we need to show that there are no more than n/2 edges that can be coloured the same in an edge coloring of G, we need to optimize the edges coloured in the graph.

To optimize the edges coloured, we need the graph with as few as adjacent sides possible. We need this kind of graph because we need to color the adjacent sides of the same vertex. So, the graph that will be the best should have an even number of vertices, each pair joint by one edge.

Here, with n number of vertices we will have n/2 edges coloured the same.

Therefore we have proved the result.

The final solution is

Here n is the numbers of vertices, so we have n/2 edges colored the same