Long paths and cycles passing through specified vertices under the average degree condition (Q5964991)

Let \(G=(V,E)\) be a \(k\)-connected graph with \(k \geq 2\), and let \(x\) and \(z\) be any distinct vertices of \(G\). \textit{P. Erdős} and \textit{T. Gallai} [Acta Math. Acad. Sci. Hung. 10, 337--356 (1959; Zbl 0090.39401)] proved that \(G\) contains a cycle of length at least \(2|E(G)|/(|V(G)|-1)\), and \textit{G. Fan} [J. Comb. Theory, Ser. B 49, No. 2, 151--180 (1990; Zbl 0713.05043)] proved that \(G\) contains an \((x,z)\)-path of length at least the average degree of the vertices other than \(x\) and \(z\). In this paper, the authors generalize the above two results. They first prove that \(G\) contains an \((x,z)\)-path passing through any of its \(k-2\) specified vertices with length at least the average degree of the vertices other than \(x\) and \(z\). Then, using this result, they prove that \(G\) contains a cycle of length at least \(2|E(G)|/(|V(G)|-1)\) passing through any of its \(k-1\) specified vertices.