3,328 research outputs found
Nowhere-Zero Flow Polynomials
In this article we introduce the flow polynomial of a digraph and use it to
study nowhere-zero flows from a commutative algebraic perspective. Using
Hilbert's Nullstellensatz, we establish a relation between nowhere-zero flows
and dual flows. For planar graphs this gives a relation between nowhere-zero
flows and flows of their planar duals. It also yields an appealing proof that
every bridgeless triangulated graph has a nowhere-zero four-flow
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
A nowhere-zero -flow on a graph is a mapping from the edges of
to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in
any fixed orientation of , at each node the sum of the labels over the
edges pointing towards the node equals the sum over the edges pointing away
from the node. We show that the existence of an \emph{integral flow polynomial}
that counts nowhere-zero -flows on a graph, due to Kochol, is a consequence
of a general theory of inside-out polytopes. The same holds for flows on signed
graphs. We develop these theories, as well as the related counting theory of
nowhere-zero flows on a signed graph with values in an abelian group of odd
order. Our results are of two kinds: polynomiality or quasipolynomiality of the
flow counting functions, and reciprocity laws that interpret the evaluations of
the flow polynomials at negative integers in terms of the combinatorics of the
graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.
Enumerating Colorings, Tensions and Flows in Cell Complexes
We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex , generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in for
some ) or integral (with values in ). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory
Series
Flows on Simplicial Complexes
Given a graph , the number of nowhere-zero \ZZ_q-flows is
known to be a polynomial in . We extend the definition of nowhere-zero
\ZZ_q-flows to simplicial complexes of dimension greater than one,
and prove the polynomiality of the corresponding function
for certain and certain subclasses of simplicial complexes.Comment: 10 pages, to appear in Discrete Mathematics and Theoretical Computer
Science (proceedings of FPSAC'12
Bounds on the Coefficients of Tension and Flow Polynomials
The goal of this article is to obtain bounds on the coefficients of modular
and integral flow and tension polynomials of graphs. To this end we make use of
the fact that these polynomials can be realized as Ehrhart polynomials of
inside-out polytopes. Inside-out polytopes come with an associated relative
polytopal complex and, for a wide class of inside-out polytopes, we show that
this complex has a convex ear decomposition. This leads to the desired bounds
on the coefficients of these polynomials.Comment: 16 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
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