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The Dirac's theorem states that: "For a Graph G with N vertices, if the degree of each vertex is atleast N/2 then, the Graph has a Hamilton Circuit." Can the same be said if a graph has a Hamilton Circuit then the degree of each vertex is atleast N/2 ?

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No. A graph with only the edges of a polygon has an Hamilton circle, but the degree of each vertex is 2.

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    $\begingroup$ Such a graph is usually called a cycle or circuit. $\endgroup$ – Yuval Filmus Aug 7 at 7:21
  • $\begingroup$ I think that's the condition for an Euler Circuit to exist. @shuki $\endgroup$ – Shashwat Vaibhav Aug 8 at 8:04

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