isomorphism of graph

Question

I have a doubt about an example of isomorphism of graphs. I am looking at the notes of group theory by Donald Kreher. (http://www.math.mtu.edu/~kreher/ABOUTME/syllabus/GTN.pdf). Please see page 9 figure1.1 What I don't get it is why this two graphs $\Gamma_1$ and $\Gamma_2$ are isomorphic. The mapping $\theta(5)=a$ and $\theta(6)=f$. In $\Gamma_1$ $4$ has nearest neighbour $5$ and $6$ but in $\Gamma_2$ that is not matching. Similarly in $\Gamma_2$ $d(or~ 3)$ has nearest neighbour $c (or~ 2)$ and $a (or~ 5)$ which is not matching with $\Gamma_1$. Please help me to clear this doubt.

Answer 1

You are correct; there is an error. The graphs are not isomorphic.

Perhaps the easiest way to see this is that the vertex $d$, in $\Gamma_2$, is incident to four edges, whereas no vertex in $\Gamma_1$ is incident to more than three.

Answer 2

These figures are not isomorphic because each degree of vertex of first fig: is not correspondence to the other fig: