Long cycles in certain graphs of large degree (Q5926151)

Let \(G=(V, E)\) be a connected graph of order \(n\). For a vertex \(v\in V\) let \(\text{deg}(v)\) be the degree of \(v\) and for vertices \(u,v\in V\) let \(\text{dist}(u,v)\) be the distance between \(u\) and \(v\). Graph \(G\) satisfies Fan's condition if \(\min_{u,v\in V} \{\max\{\text{deg}(u),\text{deg}(v)\} \mid \text{dist}(u,v)=2\} \geq n/2\). \textit{G. Fan} [J. Comb. Theory, Ser. B 37, No. 3, 221-227 (1984; Zbl 0551.05048)] proved that every 2-connected graph with Fan condition is hamiltonian. The author introduces the notion of modified Fan condition and proves the following result: Let \(G\) be a connected graph with modified Fan condition and \(X:=\{v\in V\mid \text{deg}(v)\geq n/2\}\). If \(|X|\geq 3\) then the vertices of the block \(B\) of \(G\) containing \(X\) form a cycle. If \(G\) is 2-connected then the condition \(|X|\geq 3\) can be removed.