Related papers: On the Infinite Lucchesi-Younger Conjecture I
A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of a set of edges meeting every non-empty…
In a series of three papers we develop an end space theory for directed graphs. As for undirected graphs, the ends of a digraph are points at infinity to which its rays converge. Unlike for undirected graphs, some ends are joined by limit…
Polat generalised Menger's theorem -- the maximum number of vertex-disjoint paths between two sets $A$ and $B$ equals the minimum size of an $A$-$B$ separator -- to ends of undirected graphs. In this paper we extend Menger's theorem to ends…
In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects every dicut. Edmonds and Giles conjectured that in a weighted digraph, the minimum weight of a dicut is equal to the…
Bidirected graphs are a generalisation of directed graphs that arises in the study of undirected graphs with perfect matchings. Menger's famous theorem - the minimum size of a set separating two vertex sets $X$ and $Y$ is the same as the…
Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct…
Lov\'asz and Cherkassky discovered in the 1970s independently that if $ G $ is a finite graph with a given set $ T $ of terminal vertices such that $ G $ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct…
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 well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of…
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,…
We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree…
We prove that for every set $S$ of vertices of a directed graph $D$, the maximum number of vertices in $S$ contained in a collection of vertex-disjoint cycles in $D$ is at least the minimum size of a set of vertices that hits all cycles…
In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number…
A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices,…
The digirth of a digraph is the length of a shortest directed cycle. The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the smallest size of a partition of the vertex-set into subsets inducing acyclic subgraphs. A conjecture by…
A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions…
A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated…
The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed…
The star-comb lemma is a standard tool in infinite graph theory, which states that for every infinite set $U$ of vertices in a connected graph $G$ there exists either a subdivided infinite star in $G$ with all leaves in $U$, or an infinite…