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 $δ$ were reported by Chandran et al., Krivelevich and Yuster, and Li and Shi. However, if a graph has a small minimum degree $δ$ and a large number of vertices $n$, these upper bounds are very large, linear in $n$. Hence, one may think to look for a good parameter to replace $δ$ and decrease the upper bounds significantly. Such a natural parameter is $σ_k$. In this paper, for the rainbow connection number we prove that if $G$ is a connected graph of order $n$ with $k$ independent vertices, then $rc(G)\leq 3k\frac{n-2}{σ_k+k}+6k-4$. For the rainbow vertex-connection number, we prove that $rvc(G)\leq \frac{(4k+2k^{2})n}{σ_k+k}+5k $ if $σ_k\leq 7k$ and $σ_k\geq 8k$, and $rvc(G)\leq \frac{(\frac{38k}{9}+2k^{2})n}{σ_k+k}+5k$ if $7k<σ_k< 8k$. Examples are given showing that our bounds are much better than the existing ones, i.e., for the examples $δ$ is very small but $σ_k$ is very large, and the bounds are $rc(G)< 9k-3$ and $rvc(G)\leq 9k+2k^{2}$ or $rvc(G)\leq\frac{83k}{9}+2k^{2}$, which imply that both $rc(G)$ and $rvc(G)$ can be upper bounded by constants from our upper bounds, but linear in $n$ from the existing ones.
An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected} if any two vertices are connected by a rainbow geodesic. The (\emph{strong}) \emph{rainbow connection number} of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. As an extension, Krivelevich and Yuster proposed the…
This paper studies the rainbow connection number of the power graph $Γ_G$ of a finite group $G$. We determine the rainbow connection number of $Γ_G$ if $G$ has maximal involutions or is nilpotent, and show that the rainbow connection number of $Γ_G$ is at most three if $G$ has no maximal involutions. The rainbow connection numbers of power graphs of some nonnilpotent groups are also given.
An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre and Sousa introduced the version which involves total-colourings. Dorbec,…
An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connection of the random $r$-regular graph $G=G(n,r)$ of order $n$, where $r\ge 4$ is a constant. We prove that with probability tending to one as $n$ goes to infinity the rainbow connection of $G$ satisfies $rc(G)=O(\log n)$, which is best possible up to a hidden constant.
A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. The strong rainbow connection number $src(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices $u$ and $v$ of $G$ are connected by a rainbow path of length $d(u,v)$. I…
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 are needed in order to make $G$ rainbow connected. In this paper, we proved that $rc(G)\leq 3(n+1)/5$ for all $3$-connected graphs.
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 edges of G is called the rainbow connection number (or rainbow connectivity) rc(G) of G. We investigate sharp thresholds in the Erdős-Rényi random graph for the property "rc(G) <= r" where r is a fixed integer. It is known that for r=2, rainbow connection number 2 and diameter 2 happen essentially at the same time in random graphs. For r >= 3, we conjecture th…
In 2011, Li et al. \cite{LLL} obtained an upper bound of the strong rainbow connection number of an $r$-dimensional undirected toroidal mesh. In this paper, this bound is improved. As a result, we give a negative answer to their problem.
For a graph $G$, Chartrand et al. defined the rainbow connection number $rc(G)$ and the strong rainbow connection number $src(G)$ in "G. Charand, G.L. John, K.A. Mckeon, P. Zhang, Rainbow connection in graphs, Mathematica Bohemica, 133(1)(2008) 85-98". They raised the following conjecture: for two given positive $a$ and $b$, there exists a connected graph $G$ such that $rc(G)=a$ and $src(G)=b$ if and only if $a=b\in\{1,2\}$ or $ 3\leq a\leq b$". In this short note, we will show that the conjecture is true.
A path in a total-colored graph is called \emph{total rainbow} if its edges and internal vertices have distinct colors. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k \leq\ell$, the \emph{total rainbow $k$-connection number} of $G$, denoted by $trc_k(G)$, is the minimum number of colors used in a total coloring of $G$ to make $G$ \emph{total rainbow $k$-connected}, that is, any two vertices of $G$ are connected by $k$ internally vertex-disjoint total rainbow paths. In this paper, we study the computational complexity of total rainbow $k$-connection number of graphs. We show…
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 are connected by a rainbow path. Similarly, the \emph{strong rainbow connection number} $src(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two concepts of connectivity in graphs were introduced by Chartrand et al.~in 2008. Subsequently…
Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum number of colors that are needed to make $G$ rainbow connected is called the rainbow connection number of $G$, denoted by $rc(G)$. In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if $G$ is a connected graph, then $rc(G)\leq 2α(G)-1$. Two examples $G$ are given to show that the …
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 prove that the rainbow connection number of every 2-connected graph with n vertices is at most the ceiling of n/2. The bound is optimal.
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 $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we show that $rc(G)\leq 3$, if $|E(G)|\geq {{n-2}\choose 2}+2$, and $rc(G)\leq 4$, if $|E(G)|\geq {{n-3}\choose 2}+3$. These bounds are sharp.
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 $G$ there is a rainbow path connecting them. The rainbow connection number of $G$, denoted by $rc(G)$, is defined as the smallest number of colors by using which there is a coloring such that $G$ is rainbow connected. In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for…
A vertex-colored graph $G$ is rainbow vertex-connected if any pair of distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of $G$, denoted by $rvc(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-connected. In this paper we give a Nordhaus-Gaddum-type result of the rainbow vertex-connection number. We prove that when $G$ and $\bar{G}$ are both connected, then $2\leq rvc(G)+rvc(\bar{G})\leq n-1$. Examples are given to show that both the upper bound and the lower bound are best possible for all…
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 $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. We prove if $G$ is a connected strongly $r$-regular graph and $r\geq 600$, then $rc(G)\leq3$. Specially, there is a constant $c$ such that $rc(G)\leq c$ for any connected strongly regular graph $G$.
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 $G$ rainbow connected. In this paper we give a Nordhaus-Gaddum-type result for the rainbow connection number. We prove that if $G$ and $\bar{G}$ are both connected, then $4\leq rc(G)+rc(\bar{G})\leq n+2$. Examples are given to show that the upper bound is sharp for all $n\geq 4$, and the lower bound is sharp for all $n\geq 8$. For the rest small $n=4,5,6,7,$ we al…
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 rainbow path. The minimum $k$ for which there exists such an edge-colouring is the rainbow connection number $rc(G)$ of $G.$ Recently, Bau et al. \cite{BJJKM2018} introduced this concept with the additional requirement that the edge-colouring must be proper. %An proper edge-coloured graph is said to be \emph{properly rainbow connected} if any two distinct vertices…
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 $G$ is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper, we give a sharp upper bound that $rc(G)\leq\lceil\frac{n}{2}\rceil$ for any 2-connected graph $G$ of order $n$, which improves the results of Caro et al. to best possible.
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 connecting them. For a connected graph $G$, the \emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined as the minimum number of colors that are required to make $G$ rainbow vertex-connected. In this paper, we find all the families $\mathcal{F}$ of connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant $k_\math…
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 distinct colors. In this paper we first determine the rainbow vertex-connection number of cycle $C_n$ of order $n\geq 3$, and then, based on it, prove that for any 2-connected graph $G$, $rvc(G)\leq rvc(C_n)$, giving a tight upper bound for the rainbow vertex-connection. As a consequence, we show that for a connected graph $G$ with a block decomposition $B_1, B_…
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 diameter $d$ and give their rainbow coloring in linear time. X.Deng et al. $\cite{XD}$ give a polynomial time algorithm to compute the rainbow connection number of MOPs by the Maximal fan partition method, but only obtain a compact upper bound. J. Lauri $\cite{JL}$ proved that, for chordal outerplanar graphs given an edge-coloring, to verify whether it is ra…
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 rainbow path is called the rainbow connection number $rc(G)$ of the graph $G$. For any graph $G$, $rc(G) \ge diam(G)$. We will show that for the Erdős-Rényi random graph $G(n,p)$ close to the diameter 2 threshold, with high probability if $diam(G)=2$ then $rc(G)=2$. In fact, further strengthening this result, we will show that in the random graph process, with hig…
In this paper, we investigate rainbow connection number $rc(G)$ of bridgeless outerplanar graphs $G$ with diameter 2 or 3. We proved the following results: If $G$ has diameter $2,$ then $rc(G)=3$ for fan graphs $F_{n}$ with $n\geq 7$ or $C_5,$ otherwise $rc(G)=2;$ if $G$ has diameter $3,$ then $rc(G)\leq 4$ and the bound is sharp.
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", Chakraborty et al. proved that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-Complete, and they left 4 open questions at the end, the last one of which is the following: Suppose that we are given a graph $G$ for which we are told that $rc(G)=2$. Can we rainbow-color it in polynomial time with $o(n)$ colors ? In this paper, we settle down this question…
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 vertices have distinct colors. For a connected graph $G$, the \emph{total-rainbow connection number} of $G$, denoted by $trc(G)$, is the minimum number of colors required in a total-coloring of $G$ to make $G$ total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a No…
An edge-coloured graph $G$ is {\it rainbow connected} if any two vertices are connected by a path whose edges have distinct colours. This concept was introduced by Chartrand et al. in \cite{ch01}, and it was extended to oriented graphs by Dorbec et al. in \cite{DI}. In this paper we present some results regarding this extention, mostly for the case of circulant digraphs.
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 colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of $rad(G)$ and $η(G)$, wher…
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 vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum degree $δ$, the rainbow connection number is upper bounded by $3n/(δ+ 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(λ+ 1) + 3$ and $3n/(κ+ 1) + 3$ for rainbow connectio…
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. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colors that are needed in order to make $G$ rainbow connected. In this paper, we provide a new approach to investigate the rainbow connection number of a graph $G$ according to some constraints to its complement graph $\bar{G}$. We first derive that for a connec…
Let $G$ be a finite group, the enhanced power graph of $G$, denoted by $Γ_G^e$, is the graph with vertex set $G$ and two vertices $x,y$ are edge connected in $Γ_{G}^e$ if there exist $z\in G$ such that $x,y\in\langle z\rangle$. Let $ζ$ be a edge-coloring of $Γ_G^e$. In this article, we calculate the rainbow connection number of the enhanced power graph $Γ_G^e$.
An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connection} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In the first result of this paper we prove that computing $rc(G)$ is NP-Hard solving an open problem from \cite{Ca-Yu}. In fact, we prove that it is already NP-Complete to decide if $rc(G)=2$, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph i…
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 thi…
A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that result…
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. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colors that are needed in order to make $G$ rainbow connected. Chartrand et al. obtained that $G$ is a tree if and only if $rc(G)=m$, and it is easy to see that $G$ is not a tree if and only if $rc(G)\leq m-2$, where $m$ is the number of edge of $G$. So there is…
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 $G$ is the smallest number of colors that are needed in order to make $G$ rainbow connected. Caro et al. showed an upper bound $rc(G)\leq n-δ$ for a connected graph $G$ of order $n$ with minimum degree $δ$ in "On rainbow connection, Electron. J. Combin. 15(2008), R57". Recently, Shiermeyer gave it a generalization that $rc(G)\leq n- \frac{σ_2} 2$ in "Bounds fo…
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 graph $G$ is the rainbow connection number of $G$, denoted by $\text{rc}(G)$. A simple way to rainbow-connect a graph $G$ is to color the edges of a spanning tree with distinct colors and then re-use any of these colors to color the remaining edges of $G$. This proves that $\text{rc}(G) \le |V(G)|-1$. We ask whether there is a stronger connection between tr…
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 colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every pair of distinct vertices of $G$ is connected by a rainbow path. It is known that for every integer $k\geq 2$ deciding if a graph $G$ has $rc(G)\leq k$ is NP-Hard, and a graph $G$ with $rc(G)\leq k$ has diameter $diam(G)\leq k$. In foregoing papers, we showed that a bridgeless graph with diameter 2 has rainbo…
A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$, $P_i$ and $P_j$ have different colors. An edge-coloring of a graph $G$ is said to be a rainbow monochromatic $k$-edge-connection coloring (or $RMC_k$-coloring for short) if every two distinct vertices of $G$ are connected by at least $k$ rainbow monochromatic paths. We use $rmc_k(G)$ to denote the maximum number of colors that ensures $G$ has an $RMC_k$-coloring, and t…
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 defined to be the minimum number of colors required to color the edges of $G$ such that every two distinct vertices of $G$ are connected by at least $k$ internally disjoint rainbow paths. Fujita et. al. proposed a problem that what is the minimum constant $α>0$ such that for all 2-connected graphs $G$ on $n$ vertices, we have $rc_2(G)\leq αn$. In this paper, w…
A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. It is known that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-hard. So, it is interesting to know the best upper bound of $rc(G)$ for such a graph $G$. In this paper, we show that $rc(G)\leq 5$ if $G$ is a bridgeless graph wit…
An edge-colored graph $G$ is rainbow connected if every pair of vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of $G$ is defined to be the minimum integer $t$ such that there exists an edge-coloring of $G$ with $t$ colors that makes $G$ rainbow connected. For a graph $G$ without any cut vertex, i.e., a 2-connected graph, of order $n$, it was proved that $rc(G)\leq \lceil\frac n 2 \rceil$ and the bound is tight. In this paper, we prove that for a connected graph $G$ of order $n$ with cut vertices, $rc(G) \leq\frac{n+r-1} 2$, where…
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 are needed in order to make $G$ rainbow connected. Following an idea of Caro et al., in this paper we also investigate the rainbow connection number of dense graphs. We show that for $k\geq 2$, if $G$ is a non-complete graph of order $n$ with minimum degree $δ(G)\geq \frac{n}{2}-1+log_{k}{n}$, or minimum degree-sum $σ_{2}(G)\geq n-2+2log_{k}{n}$, then $rc(G)\…
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 that for $κ=3$ or $κ= 4$, every $κ$-connected graph $G$ on $n$ vertices with diameter $\frac{n}κ-c$ satisfies $rc(G) \leq \frac{n}κ + 15c + 18$. We also show that for every maximal planar graph $G$, $rc(G) \leq \frac{n}κ + 36$. This proves a conjecture of Li et al. for graphs with large diameter and maximal planar graphs.
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 that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.
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 $rc(G)$ ($rvc(G)$) of $G$ is the minimum integer $i$ for which there exists an $i$-edge (vertex)-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. Denote by $\mathcal{G}_d(n)$ ($\mathcal{G'}_d(n)$) the set of all graphs of order $n$ with rainbow (vertex) connection number $d$, and define $e_d(n)=\min\{e(G)\,|\, G\…
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 for which $rc(G)\leq 4$. However, they could not show that the upper bound 5 is sharp. It is known that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-hard. So, it is interesting to know the best upper bound of $rc(G)$ for such a graph $G$. In this paper, we use different way to obtain the same upper bound, and moreover, examples are given to show…
For a graph $G$, we define $σ_2(G)=min \{d(u)+d(v)| u,v\in V(G), uv\not\in E(G)\}$, or simply denoted by $σ_2$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors, which was introduced by Chartrand et al. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow edge-connected. We prove that if $G$ is a connected graph of order $n$, then $rc(G)\leq 6\frac{n-2}{σ_2+2}+7$. Moreover, the bound is seen to be tight up to additive factors b…
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 connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In this paper, we study the computational complexity of vertex-rainbow connection of graphs and prove that computing $rvc(G)$ is NP-Hard. Moreover, we show that it is already NP-Complete to decide whether $rvc(G)=2$. We also p…
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 connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. Krivelevich and Yuster proved that if $G$ is a graph of order $n$ with minimum degree $δ$, then $rvc(G)<11n/δ$. In this paper, we show that $rvc(G)\leq 3n/(δ+1)+5$ for $δ\geq \sqrt{n-1}-1$ and $n\geq 290$, while $rvc(G)\leq 4n…
A total-colored graph is a graph $G$ such that both all edges and all vertices of $G$ are colored. A path in a total-colored graph $G$ is a total rainbow path if its edges and internal vertices have distinct colors. A total-colored graph $G$ is total-rainbow connected if any two vertices of $G$ are connected by a total rainbow path of $G$. The total rainbow connection number of $G$, denoted by $trc(G)$, is defined as the smallest number of colors that are needed to make $G$ total-rainbow connected. These concepts were introduced by Liu et al. Notice that for a connected graph $G$, $2diam(G)-1\…
Let $k$ be a positive integer, and $G$ be a $k$-connected graph. An edge-coloured path is \emph{rainbow} if all of its edges have distinct colours. The \emph{rainbow $k$-connection number} of $G$, denoted by $rc_k(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by $k$ internally vertex-disjoint rainbow paths. The function $rc_k(G)$ was introduced by Chartrand, Johns, McKeon and Zhang in 2009, and has since attracted significant interest. Let $t_k(n,r)$ denote the minimum number of edges in a $k$-connected graph $G$ on $n$ vertices wit…
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 are needed in order to make $G$ rainbow connected. It was proved that computing $rc(G)$ is an NP-Hard problem, as well as that even deciding whether a graph has $rc(G)=2$ is NP-Complete. It is known that deciding whether a given edge-colored graph is rainbow connected is NP-Complete. We will prove that it is still NP-Complete even when the edge-colored graph …
A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In a previous paper we showed that it is NP-Complete to decide whether a given graph $G$ has $rvc(G)=2$. In this paper we show that for every integer $k\geq 2$, deciding whether $rvc(G)\leq k$ is NP-Hard.…
