149 research outputs found
Nowhere-zero flows on signed eulerian graphs
This paper is devoted to a detailed study of nowhere-zero flows on signed
eulerian graphs. We generalise the well-known fact about the existence of
nowhere-zero -flows in eulerian graphs by proving that every signed eulerian
graph that admits an integer nowhere-zero flow has a nowhere-zero -flow. We
also characterise signed eulerian graphs with flow number , , and , as
well as those that do not have an integer nowhere-zero flow. Finally, we
discuss the existence of nowhere-zero -flows on signed eulerian graphs for
an arbitrary abelian group~.Comment: 17 pages, 1 figur
Odd decompositions of eulerian graphs
We prove that an eulerian graph admits a decomposition into closed
trails of odd length if and only if and it contains at least pairwise
edge-disjoint odd circuits and . We conjecture that a
connected -regular graph of odd order with admits a decomposition
into odd closed trails sharing a common vertex and verify the conjecture
for . The case is crucial for determining the flow number of a
signed eulerian graph which is treated in a separate paper (arXiv:1408.1703v2).
The proof of our conjecture for is surprisingly difficult and calls for
the use of signed graphs as a convenient technical tool.Comment: 15 pages, 3 figure
Circular flow number of highly edge connected signed graphs
This paper proves that for any positive integer , every essentially
-unbalanced -edge connected signed graph has circular flow
number at most .Comment: 11 page
Classification of Indecomposable Flows of Signed Graphs
An indecomposable flow on a signed graph is a nontrivial
integral flow that cannot be decomposed into , where are
nontrivial integral flows having the same sign (both or both )
at each edge of . This paper is to classify indecomposable flows into
characteristic vectors of circuits and Eulerian cycle-trees --- a class of
signed graphs having a kind of tree structure in which all cycles can be viewed
as vertices of a tree. Moreover, each indecomposable flow other than circuit
characteristic vectors can be further decomposed into a sum of certain half
circuit characteristic vectors having the same sign at each edge. The variety
of indecomposable flows of signed graphs is much richer than that of ordinary
unsigned graphs.Comment: 36 figure
Nowhere-zero flows on signed regular graphs
We study the flow spectrum and the integer flow spectrum
of signed -regular graphs. We show that if , then or .
Furthermore, if and only if has a
-factor. If has a 1-factor, then , and for
every , there is a signed -regular graph with and does not have a 1-factor.
If is a cubic graph which has a 1-factor, then . Furthermore, the following
four statements are equivalent: (1) has a 1-factor. (2) . (3) . (4) .
There are cubic graphs whose integer flow spectrum does not contain 5 or 6, and
we construct an infinite family of bridgeless cubic graphs with integer flow
spectrum .
We show that there are signed graphs where the difference between the integer
flow number and the flow number is greater than or equal to 1, disproving a
conjecture of Raspaud and Zhu.
The paper concludes with a proof of Bouchet's 6-flow conjecture for
Kotzig-graphs.Comment: 24 pages, 4 figures; final version; to appear in European J.
Combinatoric
Shorter signed circuit covers of graphs
A signed circuit is a minimal signed graph (with respect to inclusion) that
admits a nowhere-zero flow. We show that each flow-admissible signed graph on
edges can be covered by signed circuits of total length at most
, improving a recent result of Cheng et al. [manuscript, 2015].
To obtain this improvement we prove several results on signed circuit covers of
trees of Eulerian graphs, which are connected signed graphs such that removing
all bridges results in a collection of Eulerian graphs.Comment: 18 pages, 5 figure
Shortest circuit covers of signed graphs
A shortest circuit cover of a bridgeless graph is a family of
circuits that covers every edge of and is of minimum total length. The
total length of a shortest circuit cover of is denoted by
. For ordinary graphs (graphs without sign), the subject of shortest
circuit cover is closely related to some mainstream areas, such as, Tutte's
integer flow theory, circuit double cover conjecture, Fulkerson conjecture, and
others. For signed graphs , it is proved recently by
M\'a\v{c}ajov\'a, Raspaud, Rollov\'a and \v{S}koviera that if is s-bridgeless, and if is
-edge-connected.
In this paper this result is improved as follows, where and is the negativeness of . The above upper
bound can be further reduced if is -edge-connected with even
negativeness
Orientations, lattice polytopes, and group arrangements II: Modular and integral flow polynomials of graphs
We study modular and integral flow polynomials of graphs by means of subgroup
arrangements and lattice polytopes. We introduce an Eulerian equivalence
relation on orientations, flow arrangements, and flow polytopes; and we apply
the theory of Ehrhart polynomials to obtain properties of modular and integral
flow polynomials. The emphasis is on the geometrical treatment through subgroup
arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law
for the modular flow polynomial, which gives rise to an interpretation on the
values of the modular flow polynomial at negative integers, and answers a
question by Beck and Zaslavsky.Comment: 24 pages, 2 figure
Dual complementary polynomials of graphs and combinatorial interpretation on the values of the Tutte polynomial at positive integers
We introduce a modular (integral) complementary polynomial
(\kappa_{\mathbbm z}(G;x,y)) of two variables of a graph by counting the
number of modular (integral) complementary tension-flows (CTF) of with an
orientation . We study these polynomials by further introducing a
cut-Eulerian equivalence relation on orientations and geometric structures such
as the complementary open lattice polyhedron \Delta_\textsc{ctf}(G,\epsilon),
the complementary open 0-1 polytope \Delta^+_\textsc{ctf}(G,\epsilon), and
the complementary open lattice polytopes \Delta^\rho_\textsc{ctf}(G,\epsilon)
with respect to orientations . The polynomial
(\kappa_{\mathbbm z}(G;x,y)) is a common generalization of the modular
(integral) tension polynomial (\tau_\mathbbm{z}(G,x)) and the
modular (integral) flow polynomial (\phi_\mathbbm{z}(G,y)), and
can be decomposed into a sum of product Ehrhart polynomials of complementary
open 0-1 polytopes \Delta^+_\textsc{ctf}(G,\rho). There are dual
complementary polynomials and \bar\kappa_{\mathbbm
z}(G;x,y), dual to and \kappa_{\mathbbm z} respectively, in the
sense that the lattice-point counting to the Ehrhart polynomials is taken
inside a topological sum of the dilated closed polytopes
\bar\Delta^+_\textsc{ctf}(G,\rho). It turns out that the polynomial
is Whitney's rank generating polynomial , which
gives rise to a combinatorial interpretation on the values of the Tutte
polynomial at positive integers. In particular, some special values
of \kappa_\mathbbm{z} and \bar\kappa_\mathbbm{z} ( and
) count the number of certain special kinds (of equivalence
classes) of orientations.Comment: 28 pages, 3 figure
Characteristic flows on signed graphs and short circuit covers
We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8
(1956), 13--28] stating that every integer flow can be expressed as a sum of
characteristic flows of circuits. In our generalisation, the r\^ole of circuits
is taken over by signed circuits of a signed graph which occur in two types --
either balanced circuits or pairs of disjoint unbalanced circuits connected
with a path intersecting them only at its ends. As an application of this
result we show that a signed graph admitting a nowhere-zero -flow has a
covering with signed circuits of total length at most .Comment: 9 pages, 2 figure
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