5,071 research outputs found
Binary matroids and local complementation
We introduce a binary matroid M(IAS(G)) associated with a looped simple graph
G. M(IAS(G)) classifies G up to local equivalence, and determines the
delta-matroid and isotropic system associated with G. Moreover, a parametrized
form of its Tutte polynomial yields the interlace polynomials of G.Comment: This article supersedes arXiv:1301.0293. v2: 26 pages, 2 figures. v3
- v5: 31 pages, 2 figures v6: Final prepublication versio
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
Random Metric Spaces and Universality
WWe define the notion of a random metric space and prove that with
probability one such a space is isometricto the Urysohn universal metric space.
The main technique is the study of universal and random distance matrices; we
relate the properties of metric (in particulary universal) space to the
properties of distance matrices. We show the link between those questions and
classification of the Polish spaces with measure (Gromov or metric triples) and
with the problem about S_{\infty}-invariant measures in the space of symmetric
matrices. One of the new effects -exsitence in Urysohn space so called
anarchical uniformly distributed sequences. We give examples of other
categories in which the randomness and universality coincide (graph, etc.).Comment: 38 PAGE
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