Long Paths and Cycles Through Specified Vertices in \(k\)-Connected Graphs

Let \(k\) and \(d\) be integers with \(d \geq k \geq 4\), let \(G\) be a \(k\)-connected graph with \(|V(G)| \geq 2d – 1\), and let \(x\) and \(z\) be distinct vertices of \(G\). We show that if for any nonadjacent distinct vertices \(u\) and \(v\) in \(V(G) – \{x, z\}\), at least one of \(yu\) and \(zv\) has degree greater than or equal to \(d\) in \(G\), then for any subset \(Y\) of \(V(G) – \{x, z\}\) having cardinality at most \(k – 1\), \(G\) contains a path which has \(x\) and \(z\) as its endvertices, passes through all vertices in \(Y\), and has length at least \(2d – 2\).