Question

Prove that \({w_4}\) is chromatically 3-critical .

Short answer

Above figures show that \({w_4}\) is chromatically 3-critical.

Step-by-step solution

Given information

\({w_n}\): It is obtained by adding a vertex and edges from this vertex to the original vertices in : it represents a wheel of edges of size n, and it is obtained by adding a vertex and edges from this vertex to the original vertices in \({C_n}\)

If the chromatic number of a connected graph G is k, but the chromatic number of the graph produced by deleting this edge from G is k-1, the graph is said to be chromatically k-critical.

Definition

A wheel graph is a graph produced by connecting a single universal vertex to all vertices of a cycle in the mathematical discipline of graph theory. The 1-skeleton of a (n-1)-gonal pyramid can also be defined as a wheel network with n vertices.

Solution 

Here\({W_4}\) is chromatically 3-critical

After deleting anyone edge of\({w_5}\), it should become chromatically either 3-critical or above. But not k-1 critical.

The final answer of the question is that

\({w_5}\)is not chromatically 3-critical as it does not follow the definition.