Related papers: Chen and Chv\'atal's Conjecture in tournaments

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…

For an oriented graph $D$, the inversion of $X\subseteq V(D)$ in $D$ is the graph obtained by reversing the orientation of all arcs with both ends in $X$. The inversion number $\mathrm{inv}(D)$ is the minimum number of inversions needed to…

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…

An {\sf oriented perfect path double cover} ($\rm OPPDC$) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each edge of $G_s$ lies in exactly one of the paths and each vertex of $G$…

The detour order of a graph $G$, denoted $\tau(G)$, is the order of a longest path in $G$. A partition $(A, B)$ of $V(G)$ such that $\tau(\langle A \rangle) \leq a$ and $\tau(\langle B \rangle) \leq b$ is called an $(a, b)$-partition of…

In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not\to H$. The directed path on $k+1$ vertices together with the transitive tournament…

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every…

Graph orientation is a well-studied area of graph theory. A proper orientation of a graph $G = (V,E)$ is an orientation $D$ of $E(G)$ such that for every two adjacent vertices $ v $ and $ u $, $ d^{-}_{D}(v) \neq d^{-}_{D}(u)$ where…

The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…

Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study…

Given a directed graph G=(V,E) an independent set A of the vertices V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed…

Let $D=(V,A)$ be a directed graph of order $n\geq 6$. Let $W$ be a subset of $V$ with $|W|\geq 6$. Suppose that every vertex of $W$ has degree at least $(3n-3)/2$ in $D$. Then for any integer partition $|W|=n_1+n_2$ with $n_1\geq 3$ and…

Let $D$ be an oriented graph. The inversion of a set $X$ of vertices in $D$ consists in reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by ${\rm inv}(D)$, is the minimum number of inversions…

The Erd\H{o}s-Hajnal conjecture says that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a clique or stable set of size at least $n^c$. In this paper we are concerned with the case when $H$ is a…

An edge uv in a graph \Gamma\ is directionally 2-signed (or, (2,d)-signed) by an ordered pair (a,b), a,b in {+,-}, if the label l(uv) = (a,b) from u to v, and l(vu) = (b,a) from v to u. Directionally 2-signed graphs are equivalent to…

An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer $k$, there exists a constant $c_k>0$ such that any ordered graph $G$ on $n$ vertices with the property that neither $G$ nor its…

One of the De Bruijn - Erdos theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of…

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be…

A set of $n$ points in the Euclidean plane determines at least $n$ distinct lines unless these $n$ points are collinear. In 2006, Chen and Chv\'atal asked whether the same statement holds true in general metric spaces, where the line…