On partitioning the edges of an infinite digraph into directed cycles

Nash-Williams proved that for an undirected graph $G$ the set $E(G)$ 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 $k\in \mathbb{N}$ a $k$-edge-connected digraph $D$ with $s,t\in V(D)$ such that there are no edge-disjoint $s \rightarrow t$ and $t\rightarrow s$ 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 $D$ is a finite digraph with $r\in V(D)$ then there is a spanning subdigraph $E$ of $D$ such that for every vertex $v\neq r$ the following quantities are equal: the local connectivity from $r$ to $v$ in $D$, the local connectivity from $r$ to $v$ in $E$ and the indegree of $v$ in $E$. 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 $r$-rooted digraph $D$ has a spanning subdigraph $E$ with the following property. For every $v\neq r$, $E$ contains a system $\mathcal{R}_v$ of internally disjoint $r\rightarrow v$ paths such that the ingoing edges of $v$ in $E$ are exactly the last edges of the paths in $\mathcal{R}_v$. Furthermore, the path-system $\mathcal{R}_v$ is `big' in $D$ in the Erd\H{o}s-Menger sense, i.e., one can choose from each path in $\mathcal{R}_{v}$ either an edge or an internal vertex in such a way that a resulting set separates $v$ from $r$ in $D$.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 $C_4$ (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 $n<\omega$ there exists an uncountably dichromatic digraph without directed cycles shorter than $n$. He asked if it is provable already in ZFC. We answer his question positively by constructing for every infinite cardinal $\kappa$ and $n<\omega$ a digraph of size $2^{\kappa}$ with dichromatic number at least $\kappa^{+}$ which does not contain directed cycles of length less than $n$ 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 $D=(V,A)$ be a countable digraph with $s\neq t\in V$ and let $\mathcal{M}=\bigoplus_{v\in V}\mathcal{M}_v$ be a matroid on $A$ where $\mathcal{M}_v$ is a finitary matroid on the ingoing edges of $v$. We show that there is a system of edge-disjoint $s \rightarrow t$ paths $\mathcal{P}$ such that the united edge set of the paths is $\mathcal{M}$-independent, and there is a $C \subseteq A$ consists of one edge from each element of $\mathcal{P}$ for which $\mathsf{span}_{\mathcal{M}}(C)$ covers all the $s\rightarrow t$ paths in $D$

Gomory-Hu trees of infinite graphs with finite total weight

Gomory and Hu proved that if $G$ is a finite graph with non-negative weights on its edges, then there exists a tree $T$ (called now Gomory-Hu tree) on $V(G)$ such that for all $u\neq v\in V(G)$ there is an $e\in E(T)$ such that the two components of $T-e$ determines an optimal (minimal valued) cut between $u$ an $v$ in $G$. 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 $D$ be a finite digraph, and let $V_0,\dots,V_{k-1}$ be nonempty subsets of $V(D)$. The (strong form of) Edmonds' branching theorem states thatthere are pairwise edge-disjoint spanning branchings $\mathcal{B}_0,\dots, \mathcal{B}_{k-1}$ in $D$ such that the root set of $\mathcal{B}_i$ is $V_i\ (i=0,\dots,k-1)$ if and only if for all $\varnothing \neq X\subseteq V(D)$ the number of ingoing edges of $X$ is greater than or equal to the number of sets $V_i$ disjoint from $X$. 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 $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so that every vertex has a monochromatic path to some point in $S$. We consider a related question and show that for every (finite or infinite) cardinal $\kappa>0$ there is a cardinal $\lambda_\kappa$ such that in every $\kappa$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with total size at most $\lambda_\kappa$ so that every vertex $v$ has a monochromatic path of length at most two from $K$ to $v$ or from $v$ to $S$.Comment: 5 page