227 research outputs found
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
Caterpillar dualities and regular languages
We characterize obstruction sets in caterpillar dualities in terms of regular
languages, and give a construction of the dual of a regular family of
caterpillars. We show that these duals correspond to the constraint
satisfaction problems definable by a monadic linear Datalog program with at
most one EDB per rule
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
On the average rank of LYM-sets
Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x S|r(x) = i}| are log-concave. Given so that wk − 1 < wk wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F S with cardinality |F| W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn m + 1
All maximum size two-part Sperner systems - in short
In this note we give a very short proof for the description of all maximum size two-part Sperner systems
A finite word poset : In honor of Aviezri Fraenkel on the occasion of his 70th birthday
Our word posets have �nite words of bounded length as their elements, with
the words composed from a �nite alphabet. Their partial ordering follows from the
inclusion of a word as a subsequence of another word. The elemental combinatorial
properties of such posets are established. Their automorphism groups are determined
(along with similar result for the word poset studied by Burosch, Frank and
R¨ohl [4]) and a BLYM inequality is veri�ed (via the normalized matching property)
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