On \(2\)-Factors with Quadrilaterals Containing Specified Vertices in a Bipartite Graph

Abstract

In this paper we consider the problem as follows: Given a bipartite graph \(G = (V_1, V_2; E)\) with \(|V_1| = |V_2| = n\) and a positive integer \(k\), what degree condition is sufficient to ensure that for any \(k\) distinct vertices \(v_1, v_2, \ldots, v_k\) of \(G\), \(G\) contains \(k\) independent quadrilaterals \(Q_1, Q_2, \ldots, Q_k\) such that \(v_i \in V(Q_i)\) for every \(i \in \{1, 2, \ldots, k\}\), or \(G\) has a \(2\)-factor with \(k\) independent cycles of specified lengths with respect to \(\{v_1, v_2, \ldots, v_k\}\)? We will prove that if \(d(x) + d(y) \geq \left\lceil (4n + k)/3 \right\rceil\) for each pair of nonadjacent vertices \(x \in V_1\) and \(y \in V_2\), then, for any \(k\) distinct vertices \(v_1, v_2, \ldots, v_k\) of \(G\), \(G\) contains \(k\) independent quadrilaterals \(Q_1, Q_2, \ldots, Q_k\) such that \(v_i \in V(Q_i)\) for each \(i \in \{1, \ldots, k\}\). Moreover, \(G\) has a \(2\)-factor with \(k\) cycles with respect to \(\{v_1, v_2, \ldots, v_k\}\) such that \(k – 1\) of them are quadrilaterals. We also discuss the degree conditions in the above results.