22,818 research outputs found
On -chromatic numbers of graphs having bounded sparsity parameters
An -graph is characterised by having types of arcs and types
of edges. A homomorphism of an -graph to an -graph , is a
vertex mapping that preserves adjacency, direction, and type. The
-chromatic number of , denoted by , is the minimum
value of such that there exists a homomorphism of to . The
theory of homomorphisms of -graphs have connections with graph theoretic
concepts like harmonious coloring, nowhere-zero flows; with other mathematical
topics like binary predicate logic, Coxeter groups; and has application to the
Query Evaluation Problem (QEP) in graph database.
In this article, we show that the arboricity of is bounded by a function
of but not the other way around. Additionally, we show that the
acyclic chromatic number of is bounded by a function of , a
result already known in the reverse direction. Furthermore, we prove that the
-chromatic number for the family of graphs with a maximum average degree
less than , including the subfamily of planar graphs
with girth at least , equals . This improves upon previous
findings, which proved the -chromatic number for planar graphs with
girth at least is .
It is established that the -chromatic number for the family
of partial -trees is both bounded below and above by
quadratic functions of , with the lower bound being tight when
. We prove and which improves both known lower bounds and
the former upper bound. Moreover, for the latter upper bound, to the best of
our knowledge we provide the first theoretical proof.Comment: 18 page
T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions
We present exact calculations of the zero-temperature partition function for
the -state Potts antiferromagnet (equivalently, the chromatic polynomial)
for families of arbitrarily long strip graphs of the square and triangular
lattices with width and boundary conditions that are doubly periodic or
doubly periodic with reversed orientation (i.e. of torus or Klein bottle type).
These boundary conditions have the advantage of removing edge effects. In the
limit of infinite length, we calculate the exponent of the entropy, and
determine the continuous locus where it is singular. We also give
results for toroidal strips involving ``crossing subgraphs''; these make
possible a unified treatment of torus and Klein bottle boundary conditions and
enable us to prove that for a given strip, the locus is the same for
these boundary conditions.Comment: 43 pages, latex, 4 postscript figure
Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices
We present an efficient algorithm for computing the partition function of the
q-colouring problem (chromatic polynomial) on regular two-dimensional lattice
strips. Our construction involves writing the transfer matrix as a product of
sparse matrices, each of dimension ~ 3^m, where m is the number of lattice
spacings across the strip. As a specific application, we obtain the large-q
series of the bulk, surface and corner free energies of the chromatic
polynomial. This extends the existing series for the square lattice by 32
terms, to order q^{-79}. On the triangular lattice, we verify Baxter's
analytical expression for the bulk free energy (to order q^{-40}), and we are
able to conjecture exact product formulae for the surface and corner free
energies.Comment: 17 pages. Version 2: added 4 further term to the serie
A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS
sets and generalized KS sets. We then use projective KS sets to characterize
all graphs for which the chromatic number is strictly larger than the quantum
chromatic number. Here, the quantum chromatic number is defined via a nonlocal
game based on graph coloring. We further show that from any graph with
separation between these two quantities, one can construct a classical channel
for which entanglement assistance increases the one-shot zero-error capacity.
As an example, we exhibit a new family of classical channels with an
exponential increase.Comment: 16 page
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