Related papers: Rainbow Connection Number and Radius
An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that…
An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that…
An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph…
A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a…
A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$…
For a graph $G$, we define $\sigma_2(G)=min \{d(u)+d(v)| u,v\in V(G), uv\not\in E(G)\}$, or simply denoted by $\sigma_2$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct…
A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and the graph is rainbow-connected if there is a rainbow path between each pair of its vertices. The minimum number of colors needed to rainbow-connect a…
Let $G$ be a connected graph. The notion \emph{the rainbow connection number $rc(G)$} of a graph $G$ was introduced recently by Chartrand et al. Basavaraju et al. showed that for every bridgeless graph $G$ with radius $r$, $rc(G)\leq…
An edge-coloured path is rainbow if all of its edges have distinct colours. Let $G$ be a connected graph. The rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any…
In a graph $G$ with a given edge colouring, a rainbow path is a path all of whose edges have distinct colours. The minimum number of colours required to colour the edges of $G$ so that every pair of vertices is joined by at least one…
The {\em rainbow vertex-connection number}, $rvc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its vertices such that every pair of vertices is connected by at least one path whose internal vertices have…
A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is…
A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices.…
The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al.,…
An edge-coloured path is \emph{rainbow} if all the edges have distinct colours. For a connected graph $G$, the \emph{rainbow connection number} $rc(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices…
A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, the…
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…
The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are…
An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph $G$ is the…
An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there…