Degree sum conditions and cycle that contains every vertex of a balanced subset given in balanced bipartite graphs

Abstract

Let $G=(A\cup B, E)$ be a connected balanced bipartite graph of order $2n$ and $U$ a subset of $V(G)$, with $|U\cap A|=|U\cap B|$. In this paper we prove that if $\Delta_{1,1}(S)= max\{ d(a)+ d(b): a\in S\cap A$ y $b\in S\cap B\}\geq n+1$, for every independent set $S$ of order $\frac{k(U)}{2}+1$ in $G[U]$ such that $S\cap A\neq \emptyset$ and $S\cap B\neq \emptyset$, then $G$ contains a cycle that includes every vertex of $U$, where $k(U)$ denote the minimum cardinality of a set of vertices of $G$ separating two vertices of $U$ in $G$.