Related papers: The super spanning connectivity of arrangement gra…

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$)…

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning…

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…

Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair…

Let $G$ be a nontrivial connected graph of order $n$, and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such…

Generalizing well-known results of Erd\H{o}s and Lov\'asz, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $\lambda{}(H)\geq \lceil{}\frac{k-1}{k}\lambda{}(G)\rceil$, where $\lambda{}(G)$ is the…

Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have…

For a positive integer k and a graph G, we consider proper vertex-colourings of G with k colours in which all k colours are actually used. We call such a colouring a strong k-colouring. The strong k-colour graph of G, S_k(G), is the graph…

An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if…

An edge-coloured graph $G$ is rainbow connected if there exists a rainbow path between any two vertices. A graph $G$ is said to be $k$-rainbow connected if there exists an edge-colouring of $G$ with at most $k$ colours that is rainbow…

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 $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of…

A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs…

In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.], Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with…

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…

Contraction-critical graphs came from the study of minimal counterexamples to Hadwiger's conjecture. A graph is $k$-contraction-critical if it is $k$-chromatic, but any proper minor is $(k-1)$-colorable. It is a long-standing result of…

A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint…

For a graph $G$ and for two distinct vertices $u$ and $v$, let $\kappa(u,v)$ be the maximum number of vertex-disjoint paths joining $u$ and $v$ in $G$. The average connectivity matrix of an $n$-vertex connected graph $G$, written…

Let $G$ be a simple graph of order $n\geq 2$ and let $k\in \{1,\ldots ,n-1\}$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their…