50 research outputs found
Nowhere-zero integral chains and flows in bidirected graphs
AbstractGeneral results on nowhere-zero integral chain groups are proved and then specialized to the case of flows in bidirected graphs. For instance, it is proved that every 4-connected (resp. 3-connected and balanced triangle free) bidirected graph which has at least an unbalanced circuit and a nowhere-zero flow can be provided with a nowhere-zero integral flow with absolute values less than 18 (resp. 30). This improves, for these classes of graphs, Bouchet's 216-flow theorem (J. Combin. Theory Ser. B 34 (1982), 279β292). We also approach his 6-flow conjecture by proving it for a class of 3-connected graphs. Our method is inspired by Seymour's proof of the 6-flow theorem (J. Combin. Theory Ser. B 30 (1981), 130β136), and makes use of new connectedness properties of signed graphs
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.
Flows on Bidirected Graphs
The study of nowhere-zero flows began with a key observation of Tutte that in
planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of
k-tensions). Tutte conjectured that every graph without a cut-edge has a
nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero
6-flow.
For a graph embedded in an orientable surface of higher genus, flows are not
dual to colourings, but to local-tensions. By Seymour's theorem, every graph on
an orientable surface without the obvious obstruction has a nowhere-zero
6-local-tension. Bouchet conjectured that the same should hold true on
non-orientable surfaces. Equivalently, Bouchet conjectured that every
bidirected graph with a nowhere-zero -flow has a nowhere-zero
6-flow. Our main result establishes that every such graph has a nowhere-zero
12-flow.Comment: 24 pages, 2 figure