5,173 research outputs found
Observability of Lattice Graphs
We consider a graph observability problem: how many edge colors are needed
for an unlabeled graph so that an agent, walking from node to node, can
uniquely determine its location from just the observed color sequence of the
walk?
Specifically, let G(n,d) be an edge-colored subgraph of d-dimensional
(directed or undirected) lattice of size n^d = n * n * ... * n. We say that
G(n,d) is t-observable if an agent can uniquely determine its current position
in the graph from the color sequence of any t-dimensional walk, where the
dimension is the number of different directions spanned by the edges of the
walk. A walk in an undirected lattice G(n,d) has dimension between 1 and d, but
a directed walk can have dimension between 1 and 2d because of two different
orientations for each axis.
We derive bounds on the number of colors needed for t-observability. Our main
result is that Theta(n^(d/t)) colors are both necessary and sufficient for
t-observability of G(n,d), where d is considered a constant.
This shows an interesting dependence of graph observability on the ratio
between the dimension of the lattice and that of the walk. In particular, the
number of colors for full-dimensional walks is Theta(n^(1/2)) in the directed
case, and Theta(n) in the undirected case, independent of the lattice
dimension.
All of our results extend easily to non-square lattices: given a lattice
graph of size N = n_1 * n_2 * ... * n_d, the number of colors for
t-observability is Theta (N^(1/t))
On Packing Colorings of Distance Graphs
The {\em packing chromatic number} of a graph is the
least integer for which there exists a mapping from to
such that any two vertices of color are at distance at
least . This paper studies the packing chromatic number of infinite
distance graphs , i.e. graphs with the set of
integers as vertex set, with two distinct vertices being
adjacent if and only if . We present lower and upper bounds for
, showing that for finite , the packing
chromatic number is finite. Our main result concerns distance graphs with
for which we prove some upper bounds on their packing chromatic
numbers, the smaller ones being for :
if is odd and
if is even
Kac-Moody Symmetries of Critical Ground States
The symmetries of critical ground states of two-dimensional lattice models
are investigated. We show how mapping a critical ground state to a model of a
rough interface can be used to identify the chiral symmetry algebra of the
conformal field theory that describes its scaling limit. This is demonstrated
in the case of the six-vertex model, the three-coloring model on the honeycomb
lattice, and the four-coloring model on the square lattice. These models are
critical and they are described in the continuum by conformal field theories
whose symmetry algebras are the , , and the
Kac-Moody algebra, respectively. Our approach is based on the
Frenkel--Kac--Segal vertex operator construction of level one Kac--Moody
algebras.Comment: 42 pages, RevTex, 14 ps figures, Submitted to Nucl. Phys. B. [FS
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