On the Swap-Distances of Different Realizations of a Graphical Degree Sequence

One of the first graph theoretical problems which got serious attention (already in the fifties of the last century) was to decide whether a given integer sequence is equal to the degree sequence of a simple graph (or it is graphical for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on the swap operation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another one of the same degree sequence. This latter problem got particular emphases in connection of rapidly mixing Markov chain approaches to sample uniformly all possible realizations of a given degree sequence. (This becomes a matter of interest in connection of – among others – the study of large social networks.) Earlier there were only crude upper bounds on the shortest possible length of such swap sequences between two realizations. In this paper we develop formulae (Gallai-type identities) for these swap-distances of any two realizations of simple undirected or directed degree sequences. These identities improves considerably the known upper bounds on the swap-distances. AMS classification (2010): Primary 05C07, Secondary 05C20, 05C45