Conditions for swappability of records in a microdata set when some marginals are fixed

We consider swapping of two records in a microdata set for the purpose of disclosure control. We give some necessary and sufficient conditions that some observations can be swapped between two records under the restriction that a given set of marginals are fixed. We also give an algorithm to find another record for swapping if one wants to swap out some observations from a particular record. Our result has a close connection to the construction of Markov bases for contingency tables with given marginals.