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Let $G$ be A simple connected graph such that all degrees are at least $2$. Is it true that every vertex is on a circle ?

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It's not true. A counter example would be two disjoint cycles plus one connecting edge. The connecting edge is not part of any cycle.

Edit: To get a vertex that is not on a cylce one can modify the example above by putting an extra vertex in the middle of the connecting edge.

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  • $\begingroup$ but i meant vertices, not edges $\endgroup$ – boaz Sep 11 at 11:12
  • $\begingroup$ still, in a circle we can visit a vertex twice, so your example does not contradicts the assertion. $\endgroup$ – boaz Sep 11 at 14:11

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