Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Disjoint dijoins   A dijoin in a digraph is a set of edges meeting every directed cut. D. R.
Woodall conjectured in 1976 that if G is a digraph, and every directed cut of G
has at least k edges, then there are k pairwise disjoint dijoins. This remains
open, but a capacitated version is known to be false. In particular, A.
Schrijver gave a digraph G and a subset S of its edge-set, such that every
directed cut contains at least two edges in S, and yet there do not exist two
disjoint dijoins included in S. In Schrijver's example, G is planar, and the
subdigraph formed by the edges in S consists of three disjoint paths.
  We conjecture that when k = 2, the disconnectedness of S is crucial: more
precisely, that if G is a digraph, and S is a subset of the edges of G that
forms a connected subdigraph (as an undirected graph), and every directed cut
of G contains at least two edges in S, then we can partition S into two
dijoins.
  We prove this in two special cases: when G is planar, and when the subdigraph
formed by the edges in S is a subdivision of a caterpillar.

1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat