Related papers: Strong Arcwise Connectedness

A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $\gamma=\gamma(\G)$, such that for any two adjacent vertices $u,v$ of $\G$ there are exactly $\gamma$ vertices of $\G$ which are…

Let $G$ be a finite group. The order supergraph of $G$ is the graph with vertex set $G$, and two distinct vertices $x,y$ are adjacent if $o(x)\mid o(y)$ or $o(y)\mid o(x)$. The enhanced power graph of $G$ is the graph whose vertex set is…

Connectivity of temporal graphs has been widely studied both as graph theory and as gossip theory. In particular, it is well known that in order to connect every vertex to every other, a temporal graph needs to have at least $2n-4$ edges…

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without…

We perform a massive evaluation of neural networks with architectures corresponding to random graphs of various types. We investigate various structural and numerical properties of the graphs in relation to neural network test accuracy. We…

Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$…

Let $G$ be a graph on $n$ vertices. A vertex of $G$ with degree at least $n/2$ is called a heavy vertex, and a cycle of $G$ which contains all the heavy vertices of $G$ is called a heavy cycle. In this paper, we characterize the graphs…

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to…

General treebank analyses are graph structured, but parsers are typically restricted to tree structures for efficiency and modeling reasons. We propose a new representation and algorithm for a class of graph structures that is flexible…

We call a pair of non-adjacent vertices in G a non-edge. Contraction of a non-edge {u, v} in G is the replacement of u and v with a single vertex z and then making all the vertices that are adjacent to u or v adjacent to z. A non-edge {u,…

The {\em disjointness graph} $G=G({\cal S})$ of a set of segments ${\cal S}$ in $R^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We…

A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on…

We complement our study of 2-connectivity in directed graphs, by considering the computation of the following 2-vertex-connectivity relations: We say that two vertices v and w are 2-vertex-connected if there are two internally…

We classify the connected-homogeneous digraphs with more than one end. We further show that if their underlying undirected graph is not connected-homogeneous, they are highly-arc-transitive.

An oriented graph is called $k$-anti-traceable if the subdigraph induced by every subset with $k$ vertices has a hamiltonian anti-directed path. In this paper, we consider an anti-traceability conjecture. In particular, we confirm this…

A k-bend right-angle-crossing drawing or (k-bend RAC drawing}, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are 90 degrees. Accordingly, a graph…

In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$-isoregularity, and the $t$-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using…

We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not…

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are…

Let $G$ be a strongly connected directed graph and $u,v,w\in V(G)$ be three vertices. Then $w$ strongly resolves $u$ to $v$ if there is a shortest $u$-$w$-path containing $v$ or a shortest $w$-$v$-path containing $u$. A set $R\subseteq…