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Let $G$ be the graph on the left and $H$ be the graph on the right.

For $G$:

number of edges: $9$

number of vertices: $6$

degree sequence: $3,3,3,3,3,3$

For $H$:

number of edges: $9$

number of vertices: $6$

degree sequence: $3,3,3,3,3,3$

I am having trouble proving these two are not isomorphic. I see $4$-cycles in $H$ but not in $G$.

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  • $\begingroup$ Please exhibit a cycle of odd length in $H$. There are none. $\endgroup$
    – C Monsour
    Dec 12 '18 at 16:47
  • $\begingroup$ Was a reply to someone else's comment that they have since deleted. See my answer below. $\endgroup$
    – C Monsour
    Dec 12 '18 at 17:14
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It's pretty easy to see they are in fact isomorphic. Each is the complete bipartite graph on two sets of three vertices each: the sets being the upper and lower vertices on the left, and sets of every other vertex on the ones arranged on a hexagon.

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