Minimum Length of Cycles Through Specified Vertices in Graphs with Wide-Diameter at Most \(d\)

Let \(k\) be a positive integer and let \(G\) be a graph. For two distinct vertices \(x, y \in V(G)\), the \(k\)-wide-distance \(d_k(x, y)\) between \(x\) and \(y\) is the minimum integer \(l\) such that there exist \(k\) vertex-disjoint \((x, y)\)-paths whose lengths are at most \(l\). We define \(d_k(x, x) = 0\). The \(k\)-wide-diameter \(d_k(G)\) of \(G\) is the maximum value of the \(k\)-wide-distance between two vertices of \(G\). In this paper we show that if \(G\) is a graph with \(d_k(G) \geq 2\) (\(k \geq 3\)), then there exists a cycle which contains specified \(k\) vertices and has length at most \(2(k – 3)(\operatorname{d_k}(G) – 1) + \max\{3d_k(G), \lfloor\frac{18d_k(G)-16}{5}\rfloor \}\).