Cycles Containing Given Subsets in \(1\)-Tough Graphs

For a graph \(G = (V, E)\) and \(X \subseteq V(G)\), let \(\operatorname{dist}_G(u, v)\) be the distance between the vertices \(u\) and \(v\) in \(G\) and \(\sigma_3(X)\) denote the minimum value of the degree sum (in \(G\)) of any three pairwise non-adjacent vertices of \(X\). We obtain the main result: If \(G\) is a \(1\)-tough graph of order \(n\) and \(X \subseteq V(G)\) such that \(\sigma_3(X) \geq n\) and, for all \(x, y \in X\), \(\operatorname{dist}_G(x, y) = 2\) implies \(\max\{d(x), d(y)\} \geq \frac{n-4}{2}\), then \(G\) has a cycle \(C\) containing all vertices of \(X\). This result generalizes a result of Bauer, Broersma, and Veldiman.