  Since 1997 a considerable effort has been spent on the study of the swap
(switch) Markov chains on graphic degree sequences. Several results were proved
on rapidly mixing Markov chains on regular simple, on regular directed, on
half-regular directed and on half-regular bipartite degree sequences. In this
paper, the main result is the following: Let $U$ and $V$ be disjoint finite
sets, and let $0 < c_1 \le c_2 < |U|$ and $0 < d_1 \le d_2 < |V|$ be integers.
Furthermore, assume that the bipartite degree sequence on $U \cup V$ satisfies
$c_1 \le d(v) \le c_2\ : \ \forall v\in V$ and $d_1 \le d(u) \le d_2 \ : \
\forall u\in U$. Finally assume that $(c_2-c_1 -1)(d_2 -d_1 -1) < 1 + \max
\{c_1(|V| -d_2), d_1(|V|- c_2) \}$. Then the swap Markov chain on this
bipartite degree sequence is rapidly mixing. The technique applies on directed
degree sequences as well, with very similar parameter values.
  These results are germane to the recent results of Greenhill and Sfragara
about fast mixing MCMC processes on simple and directed degree sequences, where
the maximum degrees are $O(\sqrt{\# \mbox{ of edges}})$. The results are
somewhat comparable on directed degree sequences: while the GS results are
better applicable for degree sequences developed under some scale-free random
process, our new results are better fitted to degree sequences developed under
the Erd\H{o}s -- R\'enyi model. For example our results cover all regular
degree sequences, the GS model is not applicable when the average degree is $ >
n/16.$
