Recently, the minimal excludant of partitions has been studied extensively. In this paper, we consider the minimal excludant of bipartitions. The minimal excludant of a bipartition \(\pi =(\pi _1, \pi _2)\) refers to the smallest integer k that does not appear simultaneously in partitions \(\pi _1\) and \(\pi _2\) . Let \({{\,\textrm{mex}\,}}_2^o(n)\) (resp., \({{\,\textrm{mex}\,}}_2^e(n)\) ) be the number of bipartitions \(\pi \) of n with the minimal excludant of \(\pi \) being odd (resp., even), and \(p_{2}(n)\) be the number of bipartitions of n. We prove that \({{\,\textrm{mex}\,}}_2^o(n)>{{\,\textrm{mex}\,}}_2^e(n)\) , for \(n\ge 1\) . It is surprising that \({{\,\textrm{mex}\,}}_2^o(5n+4)\equiv {{\,\textrm{mex}\,}}_2^e(5n+4)\equiv 0\pmod 5\) , which refines the congruence \(p_{2}(5n+4)\equiv 0\pmod 5\) . We also consider three arithmetic functions related to the sum of the minimal excludants of bipartitions and establish congruences modulo 4 and 8 for two of these arithmetic functions. Finally, we propose some problems for future work.
