164 research outputs found
-Parking Functions, Acyclic Orientations and Spanning Trees
Given an undirected graph , and a designated vertex , the
notion of a -parking function (with respect to ) was independently
developed and studied by various authors, and has recently gained renewed
attention. This notion generalizes the classical notion of a parking function
associated with the complete graph. In this work, we study properties of {\em
maximum} -parking functions and provide a new bijection between them and the
set of spanning trees of with no broken circuit. As a case study, we
specialize some of our results to the graph corresponding to the discrete
-cube . We present the article in an expository self-contained form,
since we found the combinatorial aspects of -parking functions somewhat
scattered in the literature, typically treated in conjunction with sandpile
models and closely related chip-firing games.Comment: Added coauthor, extension of v2 with additional results and
references. 28 pages, 2 figure
Bigraphical Arrangements
We define the bigraphical arrangement of a graph and show that the
Pak-Stanley labels of its regions are the parking functions of a closely
related graph, thus proving conjectures of Duval, Klivans, and Martin and of
Hopkins and Perkinson. A consequence is a new proof of a bijection between
labeled graphs and regions of the Shi arrangement first given by Stanley. We
also give bounds on the number of regions of a bigraphical arrangement.Comment: Added Remark 19 addressing arbitrary G-parking functions; minor
revision
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
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