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 {\em graphical} for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on the {\em 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 fast 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 {\em swap-distance}s of any two realizations of simple undirected or directed degree sequences. These identities improves considerably the known upper bounds on the swap-distances.Comment: to be publishe

A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs

One of the simplest ways to decide whether a given finite sequence of positive integers can arise as the degree sequence of a simple graph is the greedy algorithm of Havel and Hakimi. This note extends their approach to directed graphs. It also studies cases of some simple forbidden edge-sets. Finally, it proves a result which is useful to design an MCMC algorithm to find random realizations of prescribed directed degree sequences.Comment: 11 pages, 1 figure submitted to "The Electronic Journal of Combinatorics

Magyar Siketnéma-Oktatás 09 (1907) 01

A Siketnéma-Intézeti Tanárok Országos Egyesületének hivatalos lapja 9. évfolyam, 1. szám Kecskemét, 1907. január hó. Előzménye a "Szemle a siketnémák, vakok, hülyék, gyengeelméjűek, dadogók és hebegők oktatásával foglalkozók szakközlönye". A lap a 26. évfolyam 6. számától (1924) "Siketnémák és vakok oktatásügye a Siketnémák és Vakok Tanárai Országos Egyesületének hivatalos lapja" cím alatt jelent meg