We show that for every positive integer $k$, any tournament with minimum out-degree at least $(2+o(1))k^2$ contains a subdivision of the complete directed graph on $k$ vertices, which is best possible up to a factor of $8$. This may be viewed as a directed analogue of a theorem proved by Bollobás and Thomason, and independently by Komlós and Szemerédi, concerning subdivisions of cliques in graphs with sufficiently high average degree. We also consider the following problem: given $k$, what is the smallest positive integer $f(k)$ such that any $f(k)$-vertex tournament contains a $1$-subdivision of the transitive tournament on $k$ vertices? We show that $f(k)= O\left (k^2\log^3 k\right)$ which is best possible up to the logarithmic factors.
We prove the existence of a function $h(k)$ such that every simple digraph with minimum outdegree greater than $h(k)$ contains an immersion of the transitive tournament on $k$ vertices. This solves a conjecture of Devos, McDonald, Mohar and Scheide.
Continuing the program begun in the first paper of this series, we present a pair of non-reconstructable tournament on $p$ vertices for each $p=2^n \ge 4$.
The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive tournament $H_k$ satisfies $\vec{r}(H_k)= O(k^2\log\log k)$. This is tight up to multiplicative $\log\log k$-term. We also show that if $T$ is an $n$-vertex tournament with $Δ^+(T)-δ^+(T)= O(n/k) - k^2$, then $T$ contains a $1$-subdivision of $\vec{K}_k$, a complete $k$-vertex digraph with all possible $k(k-1)$ arcs. This is also tight up to multiplicative constant.
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every $n$-vertex graph with at least $\varepsilon n^2$ edges contains a $1$-subdivision of the complete graph on $c_{\varepsilon}\sqrt{n}$ vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least $(2+o(1))k^2$ vertices contains th…
Mader conjectured that for all k there is an integer d(k) such that every digraph of minimum outdegree at least d(k) contains a subdivision of a transitive tournament of order k. In this note we observe that if the minimum outdegree of a digraph is sufficiently large compared to its order then one can even guarantee a subdivision of a large complete digraph. More precisely, let G be a digraph of order n whose minimum outdegree is at least d. Then G contains a subdivision of a complete digraph of order at least d^2/(8n^{3/2}).
Seymour's distance two conjecture states that in any digraph there exists a vertex (a "Seymour vertex") that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements along lines suggested by Seymour's conjecture, proving that almost surely there are a "large" number of Seymour vertices in random tournaments and "even more" in general random digraphs.
A $\overrightarrow{P_{3}}$-decomposition of a directed graph $D$ is a partition of the arcs of $D$ into directed paths of length $2$. In this paper, we give a characterization for a tournament and a bipartite digraph admitting a $\overrightarrow{P_{3}}$-decomposition. This solves a problem posed by Diwan ($\overrightarrow{P_{3}}$-decomposition of directed graphs, Discrete Appl. Math., http:// dx.doi.org/10.1016/j.dam.2016.01.039.).
The problem of when a given digraph contains a subdivision of a fixed digraph $F$ is considered. Bang-Jensen et al. laid out foundations for approaching this problem from the algorithmic point of view. In this paper we give further support to several open conjectures and speculations about algorithmic complexity of finding $F$-subdivisions. In particular, up to 5 exceptions, we completely classify for which 4-vertex digraphs $F$, the $F$-subdivision problem is polynomial-time solvable and for which it is NP-complete. While all NP-hardness proofs are made by reduction from some version of the 2…
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to these questions (which is not quite the predicted one). Our approach is based on the recent `hypergraph containers' method, developed independently by Saxton and Thomason as well as by Balogh, Morris and Samotij. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. Along…
In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we…
We prove that for every digraph $F$ and every assignment of pairs of integers $(r_e,q_e)_{e \in A(F)}$ to its arcs there exists an integer $N$ such that every digraph $D$ with dichromatic number at least $N$ contains a subdivision of $F$ in which $e$ is subdivided into a directed path of length congruent to $r_e$ modulo $q_e$, for every $e \in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result.
The Rényi $α$-entropy $H_α$ of complete antisymmetric directed graphs (i.e., tournaments) is explored. We optimize $H_α$ when $α= 2$ and $3$, and find that as $α$ increases $H_α$'s sensitivity to what we refer to as `regularity' increases as well. A regular tournament on $n$ vertices is one with each vertex having out-degree $\frac{n-1}{2}$, but there is a lot of diversity in terms of structure among the regular tournaments; for example, a regular tournament may be such that each vertex's out-set induces a regular tournament (a doubly-regular tournament) or a transitive tournament (a rotationa…
In an earlier paper the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositi…
Given $k$ pairs of vertices $(s_i,t_i)\;(1\le i\le k)$ of a digraph $G$, how can we test whether there exist vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but in an earlier paper we proved that for all fixed $k$, there is a polynomial-time algorithm to solve the problem if $G$ is a tournament (or more generally, a semicomplete digraph). Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $V(G)$ is partitioned into a bounded number of sets each inducing a semicomple…
A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments, strong digraphs, and acyclic digraphs by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.
Given $k$ pairs of vertices $(s_i,t_i)$, $1\le i\le k$, of a digraph $G$, how can we test whether there exist $k$ vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but for $k=2$ there is a polynomial-time algorithm when $G$ is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen. Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $G$ is semicomplete.
Decomposing a digraph into subdigraphs with a fixed structure or property is a classical problem in graph theory and a useful tool in a number of applications of networks and communication. A digraph is strongly connected if it contains a directed path from each vertex to all others. In this paper we consider multipartite tournaments, and we study the existence of a partition of a multipartite tournament with $c$ partite sets into strongly connected $c$-tournaments. This is a continuation of the study started in 1999 by Volkmann of the existence of strongly connected subtournaments in multipar…
A $k$-majority digraph is a directed graph created by combining $k$ individual rankings on the same ground set to form a consensus where edges point in the direction indicated by a strict majority of the rankings. The $k$-majority digraph is used to model voting scenarios, where the vertices correspond to options ranked by $k$ voters. When $k$ is odd, the resulting digraph is always a tournament, called $k$-majority tournament. Let $f_k(n)$ be the minimum, over all $k$-majority tournaments with $n$ vertices, of the maximum order of an induced transitive sub-tournament. Recently, Milans, Schrei…
Let $G$ be a graph of order $n$. It is well-known that $α(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $α(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ α(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $α(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $χ(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(Δ^+(D),Δ^-(D))$. We give a short and simple proof for…
Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $ρ: A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0, \ldots , n-1 \}$, we say that there is an obstruction on $x_{i}$ whenever $(ρ(x_{i-1}, x_{i}), ρ(x_{i}, x_{i+1})) \notin A(H)$ (when $x_{0} = x_{n}$ the indices are taken modulo $n$). We denote by $O_{H}(W)$ the set $\{ i \in \{0, \ldots , n-1 \} :$ there is an obstruction on $x_{i} \}$. The $H$-length of $W$, denoted by $l_{H}(W)$, is defined by $|O_{H}(W)|+1$ whe…
We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vecχ}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $\text{mader}_{\vecχ}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is tha…
In 1995, Stiebitz asked the following question: For any positive integers $s,t$, is there a finite integer $f(s,t)$ such that every digraph $D$ with minimum out-degree at least $f(s,t)$ admits a bipartition $(A, B)$ such that $A$ induces a subdigraph with minimum out-degree at least $s$ and $B$ induces a subdigraph with minimum out-degree at least $t$? We give an affirmative answer for tournaments, multipartite tournaments, and digraphs with bounded maximum in-degrees. In particular, we show that for every $ε$ with $0<ε<1/2$, there exists an integer $δ_0$ such that every tournament with minimu…
We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs.
A digraph is {\bf eulerian} if it is connected and every vertex has its in-degree equal to its out-degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. A digraph is {\bf semicomplete} if it has no pair of non-adjacent vertices. A {\bf tournament} is a semicomplete digraph without directed cycles of length 2. Fraise and Thomassen \cite{fraisseGC3} proved that every $(k+1)$-strong tournament has a hamiltonian cycle which avoids any prescribed set of $k$ arcs. In \cite{bangsupereuler} the authors demonstrated that a number of results concerning vertex-c…
A $(2+1)$-bispindle $B(k_1,k_2;k_3)$ is the union of two $xy$-dipaths of respective lengths $k_1$ and $k_2$, and one $yx$-dipath of length $k_3$, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. conjectured that, for every positive integers $k_1, k_2, k_3$, there is an integer $g(k_1, k_2, k_3)$ such that every strongly connected digraph not containing subdivisions of $B(k_1, k_2; k_3)$ has a chromatic number at most $g(k_1, k_2, k_3)$, and they proved it only for the case where $k_2=1$. For Hamiltonian digraphs, we prove Cohen et al.'s conjecture, namely $g(k_1, k_…
The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the celebrated Erdős--Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph $H$ is a superhero if every $H$-free digraph $D$ has chromatic number bounded by a function of $H$ and $α(D)$, the independence number of the underlying graph of $D$. We prove here that a digraph is a super…
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops with a coloring of its arcs $c:A(D) \rightarrow V(H)$ ($D$ is said to be an $H$-colored digraph). A directed path $W$ in $D$ is said to be an $H$-path if and only if the consecutive colors encountered on $W$ form a directed walk in $H$. A subset $N$ of vertices of $D$ is said to be an $H$-kernel if (1) for every pair of different vertices in $N$ there is no $H$-path between them and (2) for every vertex $u$ in V($D$)$\setminus$$N$ there exists an $H$-path in $D$ from $u$ to $N$. Under this definition an $H$-kernel is a ke…
For two integers $n\geq 3$ and $2\leq p\leq n$, we denote $D(n,p)$ the digraph obtained from a directed $n$-cycle by changing the orientations of $p-1$ consecutive arcs. In this paper, we show that a family of $k$-regular $(k\geq 3)$ bipartite tournament $BT_{4k}$ contains $D(4k,p)$ for all $2\leq p\leq 4k$ unless $BT_{4k}$ is isomorphic to a digraph $D$ such that $(1,2,3,...,4k,1)$ is a Hamilton cycle and $(4m+i-1,i)\in A(D)$ and $(i,4m+i+1)\in A(D)$, where $1\leq m\leq k-1$.
K-Arc-Strong Orientations Of Semicomplete Digraphs
School of Mathematics, Shandong University（山东大学数学学院）；Department of Mathematics and Computer Science, University of Southern Denmark（南丹麦大学数学与计算机科学系）；School of Mathematics and Statistics, Ningxia University（宁夏大学数学与统计学院）
We say a digraph $G$ is a {\em minor} of a digraph $H$ if $G$ can be obtained from a subdigraph of $H$ by repeatedly contracting a strongly-connected subdigraph to a vertex. Here, we show the class of all tournaments is a well-quasi-order under minor containment.
Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments
Center for Combinatorics and LPMC, Nankai University（南开大学组合数学中心与LPMC）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院与LPMC）；School of Science, Hebei University of Technology（河北工业大学理学院）；Department of Mathematics and Computer Science, University of Southern Denmark（南丹麦大学数学与计算机系）；Department of Mathematics, University of Johannesburg（约翰内斯堡大学数学系）；Shenzhen Institutes of Advanced Technology（深圳先进技术研究院）
In this paper, we compute competition indices and periods of multipartite tournaments. We first show that the competition period of an acyclic digraph $D$ is one and $ζ(D) +1$ is a sharp upper bound of the competition index of $D$ where $ζ(D)$ is the sink elimination index of $D$. Then we prove that, especially, for an acyclic $k$-partite tournament $D$, the competition index of $D$ is $ζ(D)$ or $ζ(D) +1$ for an integer $k \ge 3$. By developing useful tools to create infinitely many directed walks in a certain regular pattern from given directed walks, we show that the competition period of a …
An arc-coloured digraph $D$ is said to be \emph{rainbow connected} if for every two vertices $u$ and $v$ there is an $uv$-path all whose arcs have different colours. The minimun number of colours required to make the digraph rainbow connected is called the \emph{rainbow connection number} of $D$, denoted $\stackrel{\rightarrow}{rc}(D)$. In \cite{Dorbec} it was showed that if $T$ is a strong tournament with $n\geq 5$ vertices, then $2\leq \stackrel{\rightarrow}{rc}(T)\leq n-1$; and that for every $n$ and $k$ such that $3\leq k\leq n-1$, there exists a tournament $T$ on $n$ vertices such that $\…
We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.
Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of digraphs whose missing graph is a comb, a complete graph minus 2 independent edges, or a complete graph minus the edges of a cycle of length 5.
We study $k$-colored kernels in $m$-colored digraphs. An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that (i) from every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and (ii) for every $u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper, we prove that for every integer $k\geq 2$ there exists a $% (k+1)$-colored digraph $D$ without $k$-colored kernel and if every directed cycle of an $m$-colored digraph is monochromatic, then it has a $k$-colored …
Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$ internally disjoint paths with the same direction (resp. with different directions allowed) between $u$ and $v$ and it contains all vertices of $D$. A digraph $D$ is $k^{*}$-strongly (resp. $k^{*}$-weakly) connected if there exists a strong (resp. weak) $k^{*}$-container between any two distinct vertices. We define the strong (resp. weak) spanning connectivit…
