103 research outputs found
Edge-disjoint Hamilton cycles in graphs
In this paper we give an approximate answer to a question of Nash-Williams
from 1970: we show that for every \alpha > 0, every sufficiently large graph on
n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8
edge-disjoint Hamilton cycles. More generally, we give an asymptotically best
possible answer for the number of edge-disjoint Hamilton cycles that a graph G
with minimum degree \delta must have. We also prove an approximate version of
another long-standing conjecture of Nash-Williams: we show that for every
\alpha > 0, every (almost) regular and sufficiently large graph on n vertices
with minimum degree at least can be almost decomposed into
edge-disjoint Hamilton cycles.Comment: Minor Revisio
Primer for the algebraic geometry of sandpiles
The Abelian Sandpile Model (ASM) is a game played on a graph realizing the
dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of
this primer is to apply the theory of lattice ideals from algebraic geometry to
the Laplacian matrix, drawing out connections with the ASM. An extended summary
of the ASM and of the required algebraic geometry is provided. New results
include a characterization of graphs whose Laplacian lattice ideals are
complete intersection ideals; a new construction of arithmetically Gorenstein
ideals; a generalization to directed multigraphs of a duality theorem between
elements of the sandpile group of a graph and the graph's superstable
configurations (parking functions); and a characterization of the top Betti
number of the minimal free resolution of the Laplacian lattice ideal as the
number of elements of the sandpile group of least degree. A characterization of
all the Betti numbers is conjectured.Comment: 45 pages, 14 figures. v2: corrected typo
On the number of spanning trees in random regular graphs
Let be a fixed integer. We give an asympotic formula for the
expected number of spanning trees in a uniformly random -regular graph with
vertices. (The asymptotics are as , restricted to even if
is odd.) We also obtain the asymptotic distribution of the number of
spanning trees in a uniformly random cubic graph, and conjecture that the
corresponding result holds for arbitrary (fixed) . Numerical evidence is
presented which supports our conjecture.Comment: 26 pages, 1 figure. To appear in the Electronic Journal of
Combinatorics. This version addresses referee's comment
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