1,381 research outputs found
On partitioning the edges of an infinite digraph into directed cycles
Nash-Williams proved that for an undirected graph the set can
be partitioned into cycles if and only if every cut has either even or infinite
number of edges. Later C. Thomassen gave a simpler proof for this and
conjectured the following directed analogue of the theorem: the edge set of a
digraph can be partitioned into directed cycles if and only if for each subset
of the vertices the cardinality of the ingoing and the outgoing edges are
equal. The aim of the paper is to prove this conjecture
Highly connected infinite digraphs without edge-disjoint back and forth paths between a certain vertex pair
We construct for all a -edge-connected digraph
with such that there are no edge-disjoint
and paths. We use in our construction "self-similar" graphs
which technique could be useful in other problems as well.Comment: 4 page
Vertex-flames in countable rooted digraphs preserving an Erd\H{o}s-Menger separation for each vertex
It follows from a theorem of Lov\'asz that if is a finite digraph with
then there is a spanning subdigraph of such that for
every vertex the following quantities are equal: the local
connectivity from to in , the local connectivity from to
in and the indegree of in .
In infinite combinatorics cardinality is often an overly rough measure to
obtain deep results and it is more fruitful to capture structural properties
instead of just equalities between certain quantities. The best known example
for such a result is the generalization of Menger's theorem to infinite
digraphs. We generalize the result of Lov\'asz above in this spirit. Our main
result is that every countable -rooted digraph has a spanning
subdigraph with the following property. For every ,
contains a system of internally disjoint
paths such that the ingoing edges of in are exactly the last edges
of the paths in . Furthermore, the path-system is `big' in in the Erd\H{o}s-Menger sense, i.e., one can choose from
each path in either an edge or an internal vertex in such a
way that a resulting set separates from in .Comment: minor non-mathematical change
The Complete Lattice of Erd\H{o}s-Menger Separations
F. Escalante and T. Gallai studied in the seventies the structure of
different kind of separations and cuts between a vertex pair in a (possibly
infinite) graph. One of their results is that if there is a finite separation,
then the optimal (i.e. minimal sized) separations form a finite distributive
lattice with respect to a natural partial order. Furthermore, any finite
distributive lattice can be represented this way.
If there is no finite separation then cardinality is a too rough measure to
capture being 'optimal'. Menger's theorem provides a structural
characterization of optimality if there is a finite separation. We use this
characterization to define Erd\H{o}s-Menger separations even if there is no
finite separation. The generalization of Menger's theorem to infinite graphs
(which was not available until 2009) ensures that Erd\H{o}s-Menger separations
always exist. We show that they form a complete lattice with respect to the
partial order given by Escalante and every complete lattice can be represented
this way.Comment: 5 page
Independent and maximal branching packing in infinite matroid-rooted digraphs
We prove a common generalization of the maximal independent arborescence
packing theorem of Cs. Kir\'aly and two of our earlier works about packing
branchings in infinite digraphs.Comment: 26 page
Uncountable dichromatic number without short directed cycles
A. Hajnal and P. Erd\H{o}s proved that a graph with uncountable chromatic
number cannot avoid short cycles, it must contain for example (among
other obligatory subgraphs). It was shown recently by D. T. Soukup that, in
contrast of the undirected case, it is consistent that for any
there exists an uncountably dichromatic digraph without directed cycles shorter
than . He asked if it is provable already in ZFC. We answer his question
positively by constructing for every infinite cardinal and a digraph of size with dichromatic number at least which does not contain directed cycles of length less than
as a subdigraph.Comment: 3 pages, 1 figur
Countable Menger theorem with finitary matroid constraints on the ingoing edges
We present a strengthening of the countable Menger theorem (edge version) of
R. Aharoni. Let be a countable digraph with and
let be a matroid on where
is a finitary matroid on the ingoing edges of . We show
that there is a system of edge-disjoint paths such that the united edge set of the paths is -independent,
and there is a consists of one edge from each element of for which covers all the paths in
Gomory-Hu trees of infinite graphs with finite total weight
Gomory and Hu proved that if is a finite graph with non-negative
weights on its edges, then there exists a tree (called now Gomory-Hu
tree) on such that for all there is an such that the two components of determines an optimal (minimal
valued) cut between an in . In this paper we extend their result to infinite weighted graphs with
finite total weight. Furthermore, we show by an example that one can not omit
the condition of finiteness of the total weight
Edmonds' Branching Theorem in Digraphs without Forward-infinite Paths
Let be a finite digraph, and let be nonempty
subsets of . The (strong form of) Edmonds' branching theorem states
thatthere are pairwise edge-disjoint spanning branchings in such that the root set of is if and only if for all the number of ingoing edges of is greater than or equal to the
number of sets disjoint from . As was shown by R. Aharoni and C.
Thomassen in \cite{aharoni1989infinite}, this theorem does not remain true for
infinite digraphs. Thomassen also proved that for the class of digraphs without
backward-infinite paths, the above theorem of Edmonds remains true. Our main
result is that for digraphs without forward-infinite paths, Edmonds' branching
theorem remains true as well
King-serf duo by monochromatic paths in k-edge-coloured tournaments
An open conjecture of Erd\H{o}s states that for every positive integer
there is a (least) positive integer so that whenever a tournament has
its edges colored with colors, there exists a set of at most
vertices so that every vertex has a monochromatic path to some point in . We
consider a related question and show that for every (finite or infinite)
cardinal there is a cardinal such that in every
-edge-coloured tournament there exist disjoint vertex sets with
total size at most so that every vertex has a
monochromatic path of length at most two from to or from to .Comment: 5 page
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