부산대학교 도서관

Summary: ``For a graph $G=(V,E)$ and $X\subseteq V(G)$, let ${\rm dist}_G(u,v)$ 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 nonadjacent 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 $x,y\in X$, ${\rm dist}_G(x,y)=2$ 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 Veldman.''