Related papers: On splitting digraphs

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

We prove that there exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integer $k$, if $T$ is a strongly $4k$-connected tournament with minimum out-degree at least $f(k)$, then $T$ is $k$-linked. This makes…

A digraph $D$ is $k$-linked if for every $2k$-tuple $ x_1,\ldots , x_k, y_1, \ldots , y_k$ of distinct vertices in $D$, there exist $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ starts at $x_i$ and ends at $y_i$,…

Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our…

Galvin showed that for all fixed $\delta$ and sufficiently large $n$, the $n$-vertex graph with minimum degree $\delta$ that admits the most independent sets is the complete bipartite graph $K_{\delta,n-\delta}$. He conjectured that except…

Judicious partitioning problems on graphs ask for partitions that bound several quantities simultaneously, which have received a lot of attentions lately. Scott asked the following natural question: What is the maximum constant $c_d$ such…

An out-branching $B^+_u$ (in-branching $B^-_u$) in a digraph $D$ is a connected spanning subdigraph of $D$ in which every vertex except the vertex $u$, called the root, has in-degree (out-degree) one. It is well-known that there exists a…

The bipartite-hole-number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, is the minimum integer $k$ such that there exist positive integers $s$ and $t$ with $s + t = k + 1$, satisfying the property that for any two disjoint sets $A, B…

Let $D$ be a digraph. We define the minimum semi-degree of $D$ as $\delta^{0}(D) := \min \{\delta^{+}(D), \delta^{-}(D)\}$. Let $k$ be a positive integer, and let $S = \{s\}$ and $T = \{t_{1}, \dots ,t_{k}\}$ be any two disjoint subsets of…

In this paper, we prove that every graph with average degree at least $s+t+2$ has a vertex partition into two parts, such that one part has average degree at least $s$, and the other part has average degree at least $t$. This solves a…

We show that, for any graph $F$ and $\eta>0$, there exists a $d_0=d_0(F,\eta)$ such that every $n$-vertex $d$-regular graph with $d \geq d_0$ has a collection of vertex-disjoint $F$-subdivisions covering at least $(1-\eta)n$ vertices. This…

The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In…

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…

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of…

Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced…

In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V_1,...,V_t such that for all i the subtournament T[V_i]…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

Scott asked the question of determining $c_d$ such that if $D$ is a digraph with $m$ arcs and minimum outdegree $d\ge 2$ then $V(D)$ has a partition $V_1, V_2$ such that $\min\left\{e(V_1,V_2),e(V_2, V_1)\right\}\geq c_dm$, where…

We prove that for every bipartite graph $H$ and positive integer $s$, the class of $K_{s,s}$-subgraph-free graphs excluding $H$ as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem…

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise…