Related papers: A note on the minimum size of $k$-rainbow connecte…
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…
Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is called a $rainbow~tree$ if no two edges of $T$ receive…
The concept of $k$-rainbow index $rx_k(G)$ of a connected graph $G$, introduced by Chartrand, Okamoto and Zhang, is a natural generalization of the rainbow connection number. Let $t(n,k,\ell)$ denote the minimum size of a connected graph…
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.,…
We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, i.e., a path where no two edges have the same colour. The minimum number of colours required for a rainbow colouring of the…
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…
For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\geq \binom{n+1}{2}+k-1$ for a…
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,…
The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In…
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 $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$ and $v$, there…
Let $G$ be an edge-colored graph with $n$ vertices. A subgraph $H$ of $G$ is called a rainbow subgraph of $G$ if the colors of each pair of the edges in $E(H)$ are distinct. We define the minimum color degree of $G$ to be the smallest…
A tree in an edge colored graph is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers $k$, $\ell$ with $k\geq 3$, the \emph{$(k,\ell)$-rainbow index} $rx_{k,\ell}(G)$ of $G$ is the…
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 path in an edge (vertex)-colored graph $G$, where adjacent edges (vertices) may have the same color, is called a rainbow path if no pair of edges (internal vertices) of the path are colored the same. The rainbow (vertex) connection number…
The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same. We show…
A tree in an edge-colored connected graph $G$ is called \emph{a rainbow tree} if no two edges of it are assigned the same color. For a vertex subset $S\subseteq V(G)$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$. A…
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 an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges and colors in $G$, respectively. A subgraph $H$ is called rainbow if $c(H)=e(H)$. Li et al. (European J. Combin., 36 (2014), 453-459) proved that every…
An edge-colored graph is a graph in which each edge is assigned a color. Such a graph is called strongly edge-colored if each color class forms an induced matching, and called rainbow if all edges receive pairwise distinct colors. In this…