Related papers: Rainbow Connection Number and Connected Dominating…
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…
A {\it 2-rainbow domination function} of a graph $G$ is a function $f$ that assigns to each vertex a set of colors chosen from the set $\{1,2\}$, such that for any $v\in V(G)$, $f(v)=\emptyset$ implies $\bigcup_{u\in N(v)}f(u)=\{1,2\}$. 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…
Let $G$ be a connected graph. The \emph{rainbow connection number $rc(G)$} of a graph $G$ was recently introduced by Chartrand et al. Li et al. proved that for every bridgeless graph $G$ with diameter 2, $rc(G)\leq 5$. They gave examples…
The rainbow connection number $rc(G)$ and the rainbow vertex-connection number $rvc(G)$ of a graph $G$ were introduced by Chartrand et al. and Krivelevich and Yuster, respectively. Good upper bounds in terms of minimum degree $\delta$ were…
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…
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 $G$, denoted $rc(G)$, is the minimum number of colors that are used to make $G$…
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 tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. A {\em $k$-rainbow coloring}of $G$ is an edge coloring of $G$ having the property that for every set $S$ of $k$…
Let $G = (V, E)$ be a graph on $n$ vertices, and let $c: E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011,…
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.…
A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a…
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…
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…
An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are…
A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path…
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $\delta^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in…
A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal…
A path in an edge-colored graph is said to be rainbow if no color repeats on it. An edge-colored graph is said to be rainbow $k$-connected if every pair of vertices is connected by $k$ internally disjoint rainbow paths. The rainbow…
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…