Related papers: The generalized 3-connectivity of random graphs

A \emph{vertex separator} of a connected graph $G$ is a set of vertices removing which will result in two or more connected components and a \emph{minimum vertex separator} is a set which contains the minimum number of such vertices, i.e.,…

This paper studies the propagation connectivity of a random hypergraph $\mathbb{G}$ containing both 2-edges and 3-hyperedges. We find an exact threshold of the propagation connectivity of $\mathbb{G}$: If $I_{\epsilon,r}<-1$, then…

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…

A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected…

Let $F$ and $G$ be two graphs. A spanning subgraph $H$ of $G$ is called weakly $F$-saturated if one can add to $H$ the edges of $G \setminus H$ in some order, so that whenever a new edge is added, a new copy of $F$ is formed. Obtaining…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…

The concept of pendant-tree $k$-connectivity $\tau_k(G)$ of a graph $G$, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. Let $f(n,k,\ell)$ be the minimal number of edges of a graph $G$ of order $n$ with…

Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a…

Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations…

Given a collection $\mathcal{G}=(G_1,\dots, G_h)$ of graphs on the same vertex set $V$ of size $n$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $\lambda : E(H) \rightarrow [h]$ such…

Let $\mathcal{H}=\{H_i: i<\alpha \}$ be an indexed family of graphs for some ordinal number $\alpha$. $\mathcal{H}$-decomposition of a graph $G$ is a family $\mathcal{G}=\{G_i: i<\alpha \}$ of edge-disjoint subgraphs of $G$ such that $G_i$…

For any fixed graph $G$, the subgraph isomorphism problem asks whether an $n$-vertex input graph has a subgraph isomorphic to $G$. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of…

The algebraic connectivity $a(G)$ of a graph $G$ is defined as the second smallest eigenvalue of its Laplacian matrix $L(G)$. It also admits a variational characterization as the minimum of a quadratic form associated with $L(G)$, subject…

Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that $\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there exists a path from $u$ to $v$ that uses at most one edge from each graph in…

Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so…

A path in an edge-colored graph is called a \emph{rainbow path} if all edges on it have pairwise distinct colors. For $k\geq 1$, the \emph{rainbow-$k$-connectivity} of a graph $G$, denoted $rc_k(G)$, is the minimum number of colors required…

Goodman proved that the sum of the number of triangles in a graph on $n$ nodes and its complement is at least $n^3/24$; in other words, this sum is minimized, asymptotically, by a random graph with edge density $1/2$. Erd\H{o}s conjectured…

For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if…

For a graph $G$, let $\lambda_2(G)$ denote its second smallest Laplacian eigenvalue. It was conjectured that $\lambda_2(G) + \lambda_2(\overline{G}) \geq 1$, where $\bar{G}$ is the complement of $G$. Here, we prove this conjecture in the…

We give some existence/nonexistence statements on universal graphs, which under GCH give a necessary and sufficient condition for the existence of a universal graph of size lambda with no K(kappa), namely, if either kappa is finite or…