Independent Cycles in a Bipartite Graph

Abstract

Let \(G = (V_1, V_2; E)\) be a bipartite graph with \(|V_1| = |V_2| = n \geq 2k\), where \(k\) is a positive integer. Let \(\sigma'(G) = \min\{d(u)+d(v): u\in V_1, v\in V_2, uv \not\in E(G)\}\). Suppose \(\sigma'(G) \geq 2k + 2\). In this paper, we will show that if \(n > 2k\), then \(G\) contains \(k\) independent cycles. If \(n = 2k\), then it contains \(k-1\) independent \(4\)-cycles and a \(4\)-path such that the path is independent of all the \(k-1\) \(4\)-cycles.