1,461 research outputs found
Graphs with many strong orientations
We establish mild conditions under which a possibly irregular, sparse graph
has "many" strong orientations. Given a graph on vertices, orient
each edge in either direction with probability independently. We show
that if satisfies a minimum degree condition of and has
Cheeger constant at least , then the
resulting randomly oriented directed graph is strongly connected with high
probability. This Cheeger constant bound can be replaced by an analogous
spectral condition via the Cheeger inequality. Additionally, we provide an
explicit construction to show our minimum degree condition is tight while the
Cheeger constant bound is tight up to a factor.Comment: 14 pages, 4 figures; revised version includes more background and
minor changes that better clarify the expositio
Counting degree-constrained subgraphs and orientations
The goal of this short paper to advertise the method of gauge transformations
(aka holographic reduction, reparametrization) that is well-known in
statistical physics and computer science, but less known in combinatorics. As
an application of it we give a new proof of a theorem of A. Schrijver asserting
that the number of Eulerian orientations of a --regular graph on
vertices with even is at least
. We also show that a
--regular graph with even has always at least as many Eulerian
orientations as --regular subgraphs
Asymptotic behavior of the number of Eulerian orientations of graphs
We consider the class of simple graphs with large algebraic connectivity (the
second-smallest eigenvalue of the Laplacian matrix). For this class of graphs
we determine the asymptotic behavior of the number of Eulerian orientations. In
addition, we establish some new properties of the Laplacian matrix, as well as
an estimate of a conditionality of matrices with the asymptotic diagonal
predominanceComment: arXiv admin note: text overlap with arXiv:1104.304
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
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