  Let $ B_{\ell}(n)$ denote the number of $\ell-$regular bipartitions of $n.$
In 2013, Lin \cite{Lin2013} 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 improved 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 infinities
families of congruences and multiplicative formulae for $B_2(n)$ and $B_4(n)$
by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi on
nilpotency of Hecke operator, we also find an infinite families of congruences
modulo arbitrary power of $2$ satisfied by $B_{2^{\alpha}}(n).$
