How to prove these two Graphs not isomorphic? [From Discrete Mathematics with Applications 7th Edition by Susanna Epp Chapter 10.4 Exercise 13]

Question

Here are the graphs.

The problem comes from exercise 13 of chapter 10.4 (Discrete Mathematics 7th Edition by Susanna Epp).

Could someone show me whether or not these two graphs are isomorphic? Why?

Both number of vertices in $G$ and $G^\prime$ are $8$. Both number of edges in $G$ and $G^\prime$ are $12$. Every vertex has degree $3$.

My guess is that they are not isomorphic.

Thank You.

Answer 1

It looks to me like every vertex in $G$ is on a cycle of length $5,$ but $G'$ has no cycles of length 5.

Answer 2

Another way to prove it is to show that the vertices of $G'$ can be colored with $2$ colors in such a way that adjacent vertices never get the same color, but $G$ cannot. In technical terms, the chromatic number of $G'$ is $2$, while that of $G$ is greater than $2$.