Related papers: On Rainbow Connection Number and Connectivity

A rainbow matching in an edge-coloured graph is a matching such that its edges have distinct colours. We show that every properly edge-coloured graph $G$ with $|G| \ge (9\delta(G) -5)/2$ has a rainbow matching of size $\delta(G)$, improving…

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting…

For a given graph $H$ and $n\geq 1$, let $f(n,H)$ denote the maximum number $c$ for which there is a way to color the edges of the complete graph $K_n$ with $c$ colors such that every subgraph $H$ of $K_n$ has at least two edges of the same…

Given a graph $G$ and a coloring of its edges, a subgraph of $G$ is called rainbow if its edges have distinct colors. The rainbow girth of an edge coloring of G is the minimum length of a rainbow cycle in G. A generalization of the famous…

We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette.…

A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper…

For an edge-colored graph $G$, a set $F$ of edges of $G$ is called a \emph{proper cut} if $F$ is an edge-cut of $G$ and any pair of adjacent edges in $F$ are assigned by different colors. An edge-colored graph is \emph{proper disconnected}…

A path $P$ in an edge-colored graph $G$ is called \emph{a proper path} if no two adjacent edges of $P$ are colored the same, and $G$ is \emph{proper connected} if every two vertices of $G$ are connected by a proper path in $G$. The…

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 $k$…

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with the same color. The graph $G$ is called monochromatic disconnected if any two distinct vertices of $G$ are separated by a…

Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to…

For a connected graph $G$, the \emph{rainbow connection number $rc(G)$} of a graph $G$ was introduced by Chartrand et al. In "Chakraborty et al., Hardness and algorithms for rainbow connection, J. Combin. Optim. 21(2011), 330--347",…

In this paper, we consider rainbow connection number of maximal outerplanar graphs(MOPs) on algorithmic aspect. For the (MOP) $G$, we give sufficient conditions to guarantee that $rc(G) = diam(G).$ Moreover, we produce the graph with given…

The $t$-colored rainbow saturation number $rsat_t(n,F)$ is the minimum size of a $t$-edge-colored graph on $n$ vertices that contains no rainbow copy of $F$, but the addition of any missing edge in any color creates such a rainbow copy.…

The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers…

The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the $k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this…

Let $G = (V,E)$ be an $n$-vertex graph and let $c: E \to \mathbb{N}$ be a coloring of its edges. Let $d^c(v)$ be the number of distinct colors on the edges at $v \in V$ and let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$. H. Li proved…

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…

For two integers $r\geq 2$ and $h\geq 0$, the \emph{$h$-extra $r$-component connectivity} $\kappa^h_r(G)$ of a graph $G$ is defined to be the minimum size of a subset of vertices whose removal disconnects $G$, and there are at least $r$…

The Ramsey number $r(G)$ of a graph $G$ is the minimum number $N$ such that any red-blue colouring of the edges of $K_N$ contains a monochromatic copy of $G$. Pavez-Sign\'e, Piga and Sanhueza-Matamala proved that for any function $n\leq…