552 research outputs found
The number of maximum matchings in a tree
We determine upper and lower bounds for the number of maximum matchings
(i.e., matchings of maximum cardinality) of a tree of given order.
While the trees that attain the lower bound are easily characterised, the trees
with largest number of maximum matchings show a very subtle structure. We give
a complete characterisation of these trees and derive that the number of
maximum matchings in a tree of order is at most (the
precise constant being an algebraic number of degree 14). As a corollary, we
improve on a recent result by G\'orska and Skupie\'n on the number of maximal
matchings (maximal with respect to set inclusion).Comment: 38 page
The boundary of the outer space of a free product
Let be a countable group that splits as a free product of groups of the
form , where is a finitely generated free
group. We identify the closure of the outer space
for the axes topology with the space of
projective minimal, \emph{very small} -trees, i.e. trees
whose arc stabilizers are either trivial, or cyclic, closed under taking roots,
and not conjugate into any of the 's, and whose tripod stabilizers are
trivial. Its topological dimension is equal to , and the boundary has
dimension . We also prove that any very small
-tree has at most orbits of branch points.Comment: v3: Final version, to appear in the Israel Journal of Mathematics.
Section 3, regarding the definition and properties of geometric trees, has
been rewritten to improve the exposition, following a referee's suggestio
Every tree is a large subtree of a tree that decomposes Kn or Kn,n
Let T be a tree with m edges. A well-known conjecture of Ringel states that T decomposes
the complete graph . Graham and Häggkvist conjectured that T also decomposes the complete bipartite graph . In this paper we show that there exists an integer n with n ≤[(3m - 1)/2] and a tree T₁ with n edges such that T₁ decomposes and contains T. We also show that there exists an integer n' with n' ≥ 2m-1 and a tree T₂ with n' edges such that T₂ decomposes and contains T. In the latter case, we can improve the bound if there exists a prime p such that [3m/2] ≤ p < 2m - 1.Postprint (published version
The slopes determined by n points in the plane
Let , , ..., be the slopes of the
lines connecting points in general position in the plane. The ideal
of all algebraic relations among the defines a configuration space
called the {\em slope variety of the complete graph}. We prove that is
reduced and Cohen-Macaulay, give an explicit Gr\"obner basis for it, and
compute its Hilbert series combinatorially. We proceed chiefly by studying the
associated Stanley-Reisner simplicial complex, which has an intricate recursive
structure. In addition, we are able to answer many questions about the geometry
of the slope variety by translating them into purely combinatorial problems
concerning enumeration of trees.Comment: 36 pages; final published versio
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