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Chapter 22 Graph Isomorphisms
Graphs \(G=(V(G),E(G))\) and \(H=(V(H), E(H))\) are isomorphic when there is a bijection
\begin{equation*}
\phi: V(G) \rightarrow V(H)
\end{equation*}
such that
\begin{equation*}
(u,v) \in E(G) \Longleftrightarrow (\phi(u), \phi(v)) \in E(H).
\end{equation*}
In this case, we write \(G \cong H\text{.}\)
To prove that two graphs are isomorphic, you must explicitly give the mapping between the vertices.
To prove that two graphs are not isomorphic, you must find a structural property that they do not share. Here are some structural properties: the degree sequence, being bipartite, being connected, having a \(k\)-cycle, and more.
Exercises Practice Problems
1.
Draw two 3-regular graphs on 8 vertices, one of which is connected and one of which has two components.
2.
Are the two graphs shown below isomorphic? If so, exhibit the isomorphism. If not, find a property that should be preserved by isomorphism for which the two graphs differ.
3.
Draw the seven nonisomorphic subgraphs of \(K_3\text{.}\)
4.
Prove that \(C_5 \cong \overline{C_5}\text{.}\) Can any other cycle graph be isomorphic to its complement? Justify your answer with example(s) or proof.
5.
Are the three graphs shown below isomorphic? (Are any two of them isomorphic?) If so, exhibit the isomorphism. If not, find a property that should be preserved by isomorphism for which the two graphs differ.
