102 research outputs found
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
Packing Steiner Trees
Let be a distinguished subset of vertices in a graph . A
-\emph{Steiner tree} is a subgraph of that is a tree and that spans .
Kriesell conjectured that contains pairwise edge-disjoint -Steiner
trees provided that every edge-cut of that separates has size .
When a -Steiner tree is a spanning tree and the conjecture is a
consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved
that Kriesell's conjecture holds when is replaced by , and recently
West and Wu have lowered this value to . Our main result makes a further
improvement to .Comment: 38 pages, 4 figure
Unexpected behaviour of crossing sequences
The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number
of crossings in a drawing of G on an orientable surface of genus n. We prove
that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) =
b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al.
and resolves a problem of Salazar.Comment: 21 page
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