I would like to **construct a class of $k$-connected $k$-regular  bipartite graphs with the girth at most $k-1$**.

This problem arises from a cycle. 

 - Any 2-connected 2-regular graph is a cycle, but its girth is $n$ (the number of vertices in the cycle). 
 - For a $3$-regular graph, its girth cannot be less than or equal to $2$. The same applies to  $4$-connected $4$-regular bipartite graphs, where the girth cannot be less than 3. 

Therefore, this problem may be meaningful starting from $k=5$. While I may have found some examples, I would like to ask whether there is a unified pattern for such graphs.

For example, $k=5$, we can find the following bipartite graph such that $5$-reguler $5$-regular with girth $4$. I believe such graphs are infinite.



[![enter image description here][1]][1]


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The second question is, if I limit the girth to be $k-1$ can we still find infinitely many such graphs? That is to say, 

 - **Construct a class of $k$-connected $k$-regular  bipartite graphs with the girth of $k-1$ if $k$ is odd and of $k-2$ if $k$ is even**.







  [1]: https://i.sstatic.net/FotX4.png