4,176 research outputs found
The computation of chromatic polynomials
AbstractThe planar graph representing the truncated icosahedron is a cubic graph with 60 vertices and 90 edges. The computation of the chromatic polynomial of this graph is computed by enhancing the algorithm based on the classical Delete-Contract theorem as well as finding approaches for substantially modifying a computation tree during computation. The result itself is an interesting example of the time/space tradeoffs that are important in large computations
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
We derive some new structural results for the transfer matrix of
square-lattice Potts models with free and cylindrical boundary conditions. In
particular, we obtain explicit closed-form expressions for the dominant (at
large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as
the solution of a special one-dimensional polymer model. We also obtain the
large-q expansion of the bulk and surface (resp. corner) free energies for the
zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47}
(resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <=
m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19
Postscript figures. Also included are Mathematica files data_CYL.m and
data_FREE.m. Many changes from version 1: new material on series expansions
and their analysis, and several proofs of previously conjectured results.
Final version to be published in J. Stat. Phy
An elementary chromatic reduction for gain graphs and special hyperplane arrangements
A gain graph is a graph whose edges are labelled invertibly by "gains" from a
group. "Switching" is a transformation of gain graphs that generalizes
conjugation in a group. A "weak chromatic function" of gain graphs with gains
in a fixed group satisfies three laws: deletion-contraction for links with
neutral gain, invariance under switching, and nullity on graphs with a neutral
loop. The laws lead to the "weak chromatic group" of gain graphs, which is the
universal domain for weak chromatic functions. We find expressions, valid in
that group, for a gain graph in terms of minors without neutral-gain edges, or
with added complete neutral-gain subgraphs, that generalize the expression of
an ordinary chromatic polynomial in terms of monomials or falling factorials.
These expressions imply relations for chromatic functions of gain graphs.
We apply our relations to some special integral gain graphs including those
that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining
new evaluations of and new ways to calculate the zero-free chromatic polynomial
and the integral and modular chromatic functions of these gain graphs, hence
the characteristic polynomials and hypercubical lattice-point counting
functions of the arrangements. We also calculate the total chromatic polynomial
of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page
Is the five-flow conjecture almost false?
The number of nowhere zero Z_Q flows on a graph G can be shown to be a
polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's
five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh
that \Phi_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by
Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q
\in [5,\infty). We study the real roots of \Phi_G(Q) for a family of non-planar
cubic graphs known as generalised Petersen graphs G(m,k). We show that the
modified conjecture on real flow roots is also false, by exhibiting infinitely
many real flow roots Q>5 within the class G(nk,k). In particular, we compute
explicitly the flow polynomial of G(119,7), showing that it has real roots at
Q\approx 5.0000197675 and Q\approx 5.1653424423. We moreover prove that the
graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at
Q=5 as n\to\infty (in the latter case from above and below); and that
Q_c(7)\approx 5.2352605291 is an accumulation point of real zeros of the flow
polynomials for G(7n,7) as n\to\infty.Comment: 44 pages (LaTeX2e). Includes tex file, three sty files, and a
mathematica script polyG119_7.m. Many improvements from version 3, in
particular Sections 3 and 4 have been mostly re-writen, and Sections 7 and 8
have been eliminated. (This material can now be found in arXiv:1303.5210.)
Final version published in J. Combin. Theory
From the Ising and Potts models to the general graph homomorphism polynomial
In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
We also identify the smallest graphs which are not determined by their
homomorphism polynomials for and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
How proofs are prepared at Camelot
We study a design framework for robust, independently verifiable, and
workload-balanced distributed algorithms working on a common input. An
algorithm based on the framework is essentially a distributed encoding
procedure for a Reed--Solomon code, which enables (a) robustness against
byzantine failures with intrinsic error-correction and identification of failed
nodes, and (b) independent randomized verification to check the entire
computation for correctness, which takes essentially no more resources than
each node individually contributes to the computation. The framework builds on
recent Merlin--Arthur proofs of batch evaluation of Williams~[{\em Electron.\
Colloq.\ Comput.\ Complexity}, Report TR16-002, January 2016] with the
observation that {\em Merlin's magic is not needed} for batch evaluation---mere
Knights can prepare the proof, in parallel, and with intrinsic
error-correction.
The contribution of this paper is to show that in many cases the verifiable
batch evaluation framework admits algorithms that match in total resource
consumption the best known sequential algorithm for solving the problem. As our
main result, we show that the -cliques in an -vertex graph can be counted
{\em and} verified in per-node time and space on
compute nodes, for any constant and
positive integer divisible by , where is the
exponent of matrix multiplication. This matches in total running time the best
known sequential algorithm, due to Ne{\v{s}}et{\v{r}}il and Poljak [{\em
Comment.~Math.~Univ.~Carolin.}~26 (1985) 415--419], and considerably improves
its space usage and parallelizability. Further results include novel algorithms
for counting triangles in sparse graphs, computing the chromatic polynomial of
a graph, and computing the Tutte polynomial of a graph.Comment: 42 p
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