46 research outputs found

### The generalized 3-connectivity of Cartesian product graphs

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 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 of distinct integers $i,j$, where $1\leq i,j\leq r$. For an integer
$k$ with $2\leq k\leq n$, the $k$-connectivity $\kappa_k(G)$ of $G$ is the
greatest positive integer $r$ for which $G$ contains at least $r$ internally
disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$.
Obviously, $\kappa_2(G)=\kappa(G)$ is the connectivity of $G$. Sabidussi showed
that $\kappa(G\Box H) \geq \kappa(G)+\kappa(H)$ for any two connected graphs
$G$ and $H$. In this paper, we first study the 3-connectivity of the Cartesian
product of a graph $G$ and a tree $T$, and show that $(i)$ if
$\kappa_3(G)=\kappa(G)\geq 1$, then $\kappa_3(G\Box T)\geq \kappa_3(G)$; $(ii)$
if $1\leq \kappa_3(G)< \kappa(G)$, then $\kappa_3(G\Box T)\geq \kappa_3(G)+1$.
Furthermore, for any two connected graphs $G$ and $H$ with
$\kappa_3(G)\geq\kappa_3(H)$, if $\kappa(G)>\kappa_3(G)$, then $\kappa_3(G\Box
H)\geq \kappa_3(G)+\kappa_3(H)$; if $\kappa(G)=\kappa_3(G)$, then
$\kappa_3(G\Box H)\geq \kappa_3(G)+\kappa_3(H)-1$. Our result could be seen as
a generalization of Sabidussi's result. Moreover, all the bounds are sharp.Comment: 17 page

### Note on minimally $k$-rainbow connected graphs

An edge-colored graph $G$, where adjacent edges may have the same color, is
{\it rainbow connected} if every two vertices of $G$ are connected by a path
whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if
one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$
let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected
graphs of order $n$. Schiermeyer got some exact values and upper bounds for
$t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq
d<\lceil\frac{n}{2}\rceil$. In this paper, we improve his lower bound of
$t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq
d<\lceil\frac{n}{2}\rceil$.Comment: 8 page

### The $K_{1,2}$-structure-connectivity of graphs

In this paper, we mainly investigate $K_{1,2}$-structure-connectivity for any
connected graph. Let $G$ be a connected graph with $n$ vertices, we show that
$\kappa(G; K_{1,2})$ is well-defined if $diam(G)\geq 4$, or $n\equiv 1\pmod 3$,
or $G\notin \{C_{5},K_{n}\}$ when $n\equiv 2\pmod 3$, or there exist three
vertices $u,v,w$ such that $N_{G}(u)\cap (N_{G}(v,w)\cup\{v,w\})=\emptyset$
when $n\equiv 0\pmod 3$. Furthermore, if $G$ has $K_{1,2}$-structure-cut, we
prove $\kappa(G)/3\leq\kappa(G; K_{1,2})\leq\kappa(G)$.Comment: 18 pages,15 figure