642 research outputs found
A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes
We present a version of the weighted cellular matrix-tree theorem that is
suitable for calculating explicit generating functions for spanning trees of
highly structured families of simplicial and cell complexes. We apply the
result to give weighted generalizations of the tree enumeration formulas of
Adin for complete colorful complexes, and of Duval, Klivans and Martin for
skeleta of hypercubes. We investigate the latter further via a logarithmic
generating function for weighted tree enumeration, and derive another
tree-counting formula using the unsigned Euler characteristics of skeleta of a
hypercube and the Crapo -invariant of uniform matroids.Comment: 22 pages, 2 figures. Sections 6 and 7 of previous version simplified
and condensed. Final version to appear in J. Combin. Theory Ser.
Biconed graphs, edge-rooted forests, and h-vectors of matroid complexes
A well-known conjecture of Richard Stanley posits that the -vector of the
independence complex of a matroid is a pure -sequence. The
conjecture has been established for various classes but is open for graphic
matroids. A biconed graph is a graph with two specified `coning vertices', such
that every vertex of the graph is connected to at least one coning vertex. The
class of biconed graphs includes coned graphs, Ferrers graphs, and complete
multipartite graphs. We study the -vectors of graphic matroids arising from
biconed graphs, providing a combinatorial interpretation of their entries in
terms of `edge-rooted forests' of the underlying graph. This generalizes
constructions of Kook and Lee who studied the M\"obius coinvariant (the last
nonzero entry of the -vector) of graphic matroids of complete bipartite
graphs. We show that allowing for partially edge-rooted forests gives rise to a
pure multicomplex whose face count recovers the -vector, establishing
Stanley's conjecture for this class of matroids.Comment: 15 pages, 3 figures; V2: added omitted author to metadat
Parameterized Algorithms for Directed Maximum Leaf Problems
We prove that finding a rooted subtree with at least leaves in a digraph
is a fixed parameter tractable problem. A similar result holds for finding
rooted spanning trees with many leaves in digraphs from a wide family
that includes all strong and acyclic digraphs. This settles completely an open
question of Fellows and solves another one for digraphs in . Our
algorithms are based on the following combinatorial result which can be viewed
as a generalization of many results for a `spanning tree with many leaves' in
the undirected case, and which is interesting on its own: If a digraph of order with minimum in-degree at least 3 contains a rooted
spanning tree, then contains one with at least leaves
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