The pattern of a matrix M is a (0,1)-matrix which replaces all non-zero entries of M with a 1. A directed graph is said to support M if its adjacency matrix is the pattern of M. If M is an orthogonal matrix, then a digraph which supports M must satisfy a condition known as quadrangularity. We look at quadrangularity in tournaments and determine for which orders quadrangular tournaments exist. We also look at a more restrictive necessary condition for a digraph to support an orthogonal matrix, and give a construction for tournaments which meet this condition.
The pattern of a matrix M is a (0,1)-matrix which replaces all non-zero entries of M with a 1. There are several contexts in which studying the patterns of orthogonal matrices can be useful. One necessary condition for a matrix to be orthogonal is a property known as combinatorial orthogonality. If the adjacency matrix of a directed graph forms a pattern of a combinatorially orthogonal matrix, we say the digraph is quadrangular. We look at the quadrangular property in tournaments and regular tournaments.
Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, LD, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with some elementary observations. Among the motivations of this pap…
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…
It is known that every strong tournament has directed cycles of any length, and thereby strong subtournaments of any size. In this note, we prove that they also can share a common vertex which is a king of all of them. This common vertex can be any king in the whole tournament. Further, the Hamiltonian cycles in them can be recursively constructed by inserting an additional vertex to one directed edge.
A digraph $D$ is $k$-linked if it satisfies that for every choice of disjoint sets $\{x_1,\ldots{},x_k\}$ and $\{y_1,\ldots{},y_k\}$ of vertices of $D$ there are vertex disjoint paths $P_1,\ldots{},P_k$ such that $P_i$ is an $(x_i,y_i)$-path. Confirming a conjecture by Kühn et al, Pokrovskiy proved in 2015 that every $452k$-strong tournament is $k$-linked and asked for a better linear bound. Very recently Meng et al proved that every $(40k-31)$-strong tournament is $k$-linked. In this note we use an important lemma from their paper to give a short proof that every $(13k-6)$-strong tournament o…
We consider the following problem posed by Volkmann in 2007: How close to regular must a c-partite tournament be, to secure a strongly connected subtournament of order $c$? We give sufficient conditions on the regularity of balanced $c$-partite tournaments to assure the existence of strong maximal subtournament with minimum degree at least $\left\lfloor \frac{c-2}{4}\right\rfloor+1$. We obtain this result as an application of counting the number of subtournaments of order $c$ for which a vertex has minimum out-degree (resp. in-degree) at most $q\geq 0$.
A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists $ε(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{ε(S)}$ vertices. Let $C_5$ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon-Pach-Solymosi conjecture is known to be true for the case when $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament…
The Erdős-Hajnal Conjecture states that for every $H$ there exists a constant $ε(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $|V(G)|^{ε(H)}$. The Conjecture is still open. Some time ago its directed version was formulated (see:\cite{alon}). In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. If the Conjecture is not true then the smallest counterexample is a prime tournament. For a long time the Conjecture was known only for finitely many pri…
An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon et al. states that for every tournament H there exists epsilon(H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least c|T| and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-vertex tourn…
An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon, Pack, and Solymosi states that for every tournament H there exists epsilon(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least |T|n and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-v…
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…
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.
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 $\…
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 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.
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…
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）
A fundamental property of choice functions is stability, which, loosely speaking, prescribes that choice sets are invariant under adding and removing unchosen alternatives. We provide several structural insights that improve our understanding of stable choice functions. In particular, (i) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (ii) we completely characterize which simple choice functions give rise to stable choice functions, and (iii) we prove a strong relationship between stability and a new pr…
We prove that the strong immersion order is a well-quasi-ordering on the class of semi-complete digraphs, thereby strengthening a result of Chudnovsky and Seymour that this holds for the class of tournaments.
Kostochka and Thomason independently showed that any graph with average degree $Ω(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $Ω(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger's famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_r$ minor is a digraph whose vertex set is partitioned into $r$ parts such that each part induces a strongly-connected subdigraph, and …
Let $D$ be a strongly connected digraph. The average distance $\barσ(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $ρ(D)$ and proximity $π(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $π(D)$ and $ρ(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $π(T)=ρ(T)$ if and only if …
A vertex coloring of a strong digraph $D$ is a \emph{strong vertex-monochromatic connection coloring (SVMC-coloring)} if for every pair $u, v$ of vertices in $D$ there exists an $(u,v)$-path having all its internal vertices of the same color. Let $smc_v(D)$ denote the maximum number of colors used in an SVMC-coloring of a digraph $D$. In this paper we determine the value of $smc_v(D)$, whenever $D$ is the line digraph of a digraph. Also, if $T$ is a tournament, we give conditions to find the exact value of $smc_v(T)$.
An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre and Sousa introduced the version which involves total-colourings. Dorbec,…
Asymptotics for graphically divergent series: dense digraphs and 2-SAT formulae
LaBRI, Université de Bordeaux（波尔多大学）；LIPN, Université Sorbonne Paris Nord（索邦巴黎北大学）；IMB, Université de Bourgogne（勃艮第大学）；LIB, Université de Bourgogne（勃艮第大学）；LIP6, Sorbonne Université（索邦大学）
Let $D$ be a strongly connected directed graph of order $n\geq 4$ vertices which satisfies the following condition for every triple $x,y,z$ of vertices such that $x$ and $y$ are non-adjacent: If there is no arc from $x$ to $z$, then $d(x)+d(y)+d^+(x)+d^-(z)\geq 3n-2$. If there is no arc from $z$ to $x$, then $d(x)+d(y)+d^-(x)+d^+(z)\geq 3n-2$. In \cite{[15]} (J. of Graph Theory, Vol.16, No. 5, 51-59, 1992) Y. Manoussakis proved that $D$ is Hamiltonian. In [9] it was shown that $D$ contains a pre-Hamiltonian cycle (i.e., a cycle of length $n-1$) or $n$ is even and $D$ is isomorphic to the compl…
