29,005 research outputs found
Limited packings: related vertex partitions and duality issues
A -limited packing partition (LP partition) of a graph is a
partition of into -limited packing sets. We consider the LP
partitions with minimum cardinality (with emphasis on ). The minimum
cardinality is called LP partition number of and denoted by
. This problem is the dual problem of -tuple domatic
partitioning as well as a generalization of the well-studied -distance
coloring problem in graphs.
We give the exact value of for trees and bound it for
general graphs. A section of this paper is devoted to the dual of this problem,
where we give a solution to an open problem posed in . We also revisit
the total limited packing number in this paper and prove that the problem of
computing this parameter is NP-hard even for some special families of graphs.
We give some inequalities concerning this parameter and discuss the difference
between TLP number and LP number with emphasis on trees
Boundary Partitions in Trees and Dimers
Given a finite planar graph, a grove is a spanning forest in which every
component tree contains one or more of a specified set of vertices (called
nodes) on the outer face. For the uniform measure on groves, we compute the
probabilities of the different possible node connections in a grove. These
probabilities only depend on boundary measurements of the graph and not on the
actual graph structure, i.e., the probabilities can be expressed as functions
of the pairwise electrical resistances between the nodes, or equivalently, as
functions of the Dirichlet-to-Neumann operator (or response matrix) on the
nodes. These formulae can be likened to generalizations (for spanning forests)
of Cardy's percolation crossing probabilities, and generalize Kirchhoff's
formula for the electrical resistance. Remarkably, when appropriately
normalized, the connection probabilities are in fact integer-coefficient
polynomials in the matrix entries, where the coefficients have a natural
algebraic interpretation and can be computed combinatorially. A similar
phenomenon holds in the so-called double-dimer model: connection probabilities
of boundary nodes are polynomial functions of certain boundary measurements,
and as formal polynomials, they are specializations of the grove polynomials.
Upon taking scaling limits, we show that the double-dimer connection
probabilities coincide with those of the contour lines in the Gaussian free
field with certain natural boundary conditions. These results have direct
application to connection probabilities for multiple-strand SLE_2, SLE_8, and
SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor
change
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