On strong rainbow connection number

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any two vertices $u$ and $v$ of $G$, a rainbow $u-v$ geodesic in $G$ is a rainbow $u-v$ path of length $d(u,v)$, where $d(u,v)$ is the distance between $u$ and $v$. The graph $G$ is strongly rainbow connected if there exists a rainbow $u-v$ geodesic for any two vertices $u$ and $v$ in $G$. The strong rainbow connection number of $G$, denoted $src(G)$, is the minimum number of colors that are needed in order to make $G$ strong rainbow connected. In this paper, we first investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that $G$ is a tree if and only if $src(G)=m$, we will show that $src(G)\neq m-1$, so $G$ is not a tree if and only if $src(G)\leq m-2$, where $m$ is the number of edge of $G$. Furthermore, we characterize the graphs $G$ with $src(G)=m-2$. We next give a sharp upper bound for $src(G)$ according to the number of edge-disjoint triangles in graph $G$, and give a necessary and sufficient condition for the equality.Comment: 16 page

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 kk-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

An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study