3,645 research outputs found
Quasi-isometric classification of non-geometric 3-manifold groups
We describe the quasi-isometric classification of fundamental groups of
irreducible non-geometric 3-manifolds which do not have "too many" arithmetic
hyperbolic geometric components, thus completing the quasi-isometric
classification of 3--manifold groups in all but a few exceptional cases.Comment: Minor revision (added footnote in the Introduction
Optimal-size clique transversals in chordal graphs
The following question was raised by Tuza in 1990 and Erdos et al. in 1992:
if every edge of an n-vertex chordal graph G is contained in a clique of size
at least four, does G have a clique transversal, i.e., a set of vertices
meeting all non-trivial maximal cliques, of size at most n/4? We prove that
every such graph G has a clique transversal of size at most 2(n-1)/7 if n>=5,
which is the best possible bound
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
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