Related papers: On the spanning connectivity of tournaments

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…

Given a directed acyclic graph (DAG) $G = (V,E)$, we say that $G$ is $(e,d)$-depth-robust (resp. $(e,d)$-edge-depth-robust) if for any set $S \subset V$ (resp. $S \subseteq E$) of at most $|S| \leq e$ nodes (resp. edges) the graph $G-S$…

We study a version of the lights out game played on directed graphs. For a digraph $D$, we begin with a labeling of $V(D)$ with elements of $\mathbb{Z}_k$ for $k \ge 2$. When a vertex $v$ is toggled, the labels of $v$ and any vertex that…

Let $G$ denote a graph and $k\geq2$ be an integer. A $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$-factor of $G$ is a spanning subgraph, whose every connected component is isomorphic to an element of…

Kostochka and Thomason independently showed that any graph with average degree $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result…

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq…

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough.…

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…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

An arc-colored tournament is said to be $k$-spanning for an integer $k\geq 1$ if the union of its arc-color classes of maximal valency at most $k$ is the arc set of a strongly connected digraph. It is proved that isomorphism testing of…

A graph is strongly $\Z_{\ell}$-connected if for each boundary function $\beta: V(G)\mapsto \Z_{\ell}$ with $\beta(v) \equiv d(v) \pmod{2}$ for every vertex $v$ and $\sum_{v \in V(G)} \beta(v) \equiv 0 \pmod{2\ell}$, there exists an…

Let $G = (V, E)$ be a graph with non-empty set of vertices $V$ and set of edges $E$. The \emph{eccentric connectivity index} of the graph $G$ is defined as $$\displaystyle{\xi^C(G) = \sum_{u \in V} d_u \;ecc(u)}$$ where $d_u$ is the degree…

Let $D = (V(D), A(D))$ be a digraph. A subset $S \subseteq V(D)$ is $k$-independent if the distance between every pair of vertices of $S$ is at least $k$, and it is $\ell$-absorbent if for every vertex $u$ in $V(D) \setminus S$ there exists…

In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices…

Let $T$ be a tournament with $n$ vertices $v_1,\ldots,v_n$. The skew-adjacency matrix of $T$ is the $n\times n$ zero-diagonal matrix $S_T = [s_{ij}]$ in which $s_{ij}=-s_{ji}=1$ if $ v_i $ dominates $ v_j $. We define the determinant…

For a pair of positive parameters $D,\chi$, a partition ${\cal P}$ of the vertex set $V$ of an $n$-vertex graph $G = (V,E)$ into disjoint clusters of diameter at most $D$ each is called a $(D,\chi)$ network decomposition, if the supergraph…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by fewer than $k$ other vertices. The block number $\beta(G)$ of $G$ is the largest integer $k$ such that $G$ has a $k$-block. We…

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…

The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if…

The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let $S$ be a nonempty set of vertices of $G$, a collection $\{T_1,T_2,...,T_r\}$ of trees…