2,531 research outputs found
The VC-Dimension of Graphs with Respect to k-Connected Subgraphs
We study the VC-dimension of the set system on the vertex set of some graph
which is induced by the family of its -connected subgraphs. In particular,
we give tight upper and lower bounds for the VC-dimension. Moreover, we show
that computing the VC-dimension is -complete and that it remains
-complete for split graphs and for some subclasses of planar
bipartite graphs in the cases and . On the positive side, we
observe it can be decided in linear time for graphs of bounded clique-width
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
On large bipartite graphs of diameter 3
We consider the bipartite version of the {\it degree/diameter problem},
namely, given natural numbers and , find the maximum number
of vertices in a bipartite graph of maximum degree and diameter
. In this context, the bipartite Moore bound \M^b(d,D) represents a
general upper bound for . Bipartite graphs of order \M^b(d,D) are
very rare, and determining still remains an open problem for most
pairs.
This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on
bipartite -graphs (that is, bipartite graphs of order \M^b(d,D)-4)
was carried out. Here we first present some structural properties of bipartite
-graphs, and later prove there are no bipartite -graphs.
This result implies that the known bipartite -graph is optimal, and
therefore . Our approach also bears a proof of the uniqueness of
the known bipartite -graph, and the non-existence of bipartite
-graphs.
In addition, we discover three new largest known bipartite (and also
vertex-transitive) graphs of degree 11, diameter 3 and order 190, result which
improves by 4 vertices the previous lower bound for
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