Cycles containing given subsets in 1-tough graphs. (Q2716016)

For a set \(X\) of vertices of a graph \(G\), let \(\sigma _3(X)\) denote the minimum sum of degrees of any independent triple of vertices in \(X\). The authors show that if \(G\) is a 1-tough graph of order \(n\), \(X \subset V(G)\) with \(\sigma _3(X) \geq n\), and any pair of vertices in \(X\) at distance 2 contains a vertex of degree at least \(n-4 \over 2\), then \(G\) has a cycle containing all vertices of \(X\). This generalizes a result of \textit{D.\ Bauer, H.\ J.\ Broersma} and \textit{H.\ J.\ Veldman} [Ars Comb.\ 40, 207-218 (1995; Zbl 0843.05071)].