246 research outputs found
Packing Steiner Trees
Let be a distinguished subset of vertices in a graph . A
-\emph{Steiner tree} is a subgraph of that is a tree and that spans .
Kriesell conjectured that contains pairwise edge-disjoint -Steiner
trees provided that every edge-cut of that separates has size .
When a -Steiner tree is a spanning tree and the conjecture is a
consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved
that Kriesell's conjecture holds when is replaced by , and recently
West and Wu have lowered this value to . Our main result makes a further
improvement to .Comment: 38 pages, 4 figure
The Waldschmidt constant for squarefree monomial ideals
Given a squarefree monomial ideal , we show
that , the Waldschmidt constant of , can be expressed as
the optimal solution to a linear program constructed from the primary
decomposition of . By applying results from fractional graph theory, we can
then express in terms of the fractional chromatic number of
a hypergraph also constructed from the primary decomposition of . Moreover,
expressing as the solution to a linear program enables us
to prove a Chudnovsky-like lower bound on , thus verifying a
conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree
case. As an application, we compute the Waldschmidt constant and the resurgence
for some families of squarefree monomial ideals. For example, we determine both
constants for unions of general linear subspaces of with few
components compared to , and we find the Waldschmidt constant for the
Stanley-Reisner ideal of a uniform matroid.Comment: 26 pages. This project was started at the Mathematisches
Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop "Ideals of
Linear Subspaces, Their Symbolic Powers and Waring Problems" held in February
2015. Comments are welcome. Revised version corrects some typos, updates the
references, and clarifies some hypotheses. To appear in the Journal of
Algebraic Combinatoric
N=4 Multi-Particle Mechanics, WDVV Equation and Roots
We review the relation of N=4 superconformal multi-particle models on the
real line to the WDVV equation and an associated linear equation for two
prepotentials, F and U. The superspace treatment gives another variant of the
integrability problem, which we also reformulate as a search for closed flat
Yang-Mills connections. Three- and four-particle solutions are presented. The
covector ansatz turns the WDVV equation into an algebraic condition, for which
we give a formulation in terms of partial isometries. Three ideas for
classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and
matroids. Various examples and counterexamples are displayed
Hamilton cycles in 5-connected line graphs
A conjecture of Carsten Thomassen states that every 4-connected line graph is
hamiltonian. It is known that the conjecture is true for 7-connected line
graphs. We improve this by showing that any 5-connected line graph of minimum
degree at least 6 is hamiltonian. The result extends to claw-free graphs and to
Hamilton-connectedness
A Universal Homogeneous Simple Matroid of Rank
We construct a -homogeneous universal simple matroid of rank ,
i.e. a countable simple rank~ matroid which -embeds every
finite simple rank matroid, and such that every isomorphism between finite
-subgeometries of extends to an automorphism of . We also
construct a -homogeneous matroid which is universal for the
class of finite simple rank matroids omitting a given finite projective
plane . We then prove that these structures are not -categorical,
they have the independence property, they admit a stationary independence
relation, and that their automorphism group embeds the symmetric group
. Finally, we use the free projective extension of
to conclude the existence of a countable projective plane embedding all the
finite simple matroids of rank and whose automorphism group contains
, in fact we show that
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