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Brendan McKay
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the length of cycles in a $2$-connected simple gragh

Let $G=(V,E)$ be a simple $2$-connected graph and $C$ is a cycle in $G$ satisfies:

For any vertex $v$ of $C$,there exists at least one vertex $u\in V(G)\backslash V(C)$ adjacent with $v$.

Is it true that there must exists a cycle in $G$ which is longer than $C$?

user40096
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