Let \( B_{\ell }(n)\) denote the number of \(\ell \) -regular bipartitions of n. In 2013, Lin [6] proved a density result for \(B_4(n).\) He showed that for any positive integer k, \(B_4(n)\) is almost always divisible by \(2^k.\) In this article, we significantly extend his result. We prove that \(B_{2^{\alpha }m}(n)\) and \(B_{3^{\alpha }m}(n)\) are almost always divisible by arbitrary power of 2 and 3 respectively. Further, we obtain an infinite family of congruences and internal congruences for \(B_2(n)\) and \(B_4(n)\) by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi [14] on nilpotency of Hecke operator, we prove that there exists an infinite family of congruences modulo arbitrary power of 2 satisfied by \(B_{2^{\alpha }}(n).\)
