Article

Rainbow connections of graphs: A survey

Graphs and Combinatorics (Impact Factor: 0.35). 01/2013; 29(1):1--38. DOI: 10.1007/s00373-012-1243-2
Source: DBLP

ABSTRACT 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.

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    ABSTRACT: 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 interesting problem: Characterize the graphs $G$ with $rc(G)=m-2$. In this paper, we settle down this problem. Furthermore, we also characterize the graphs $G$ with $rc(G)=m-3$.
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    ABSTRACT: 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 results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of $G$, denoted by $srvc(G)$. Observe that $rvc(G)\leq srvc(G)$ for any nontrivial connected graph $G$. In this paper, sharp upper and lower bounds of $srvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvc(G)\leq n-2$. Graphs of order $n$ such that $srvc(G)= 1, 2, n-2$ are characterized, respectively. It is also shown that, for each pair $a, b$ of integers with $a\geq 5$ and $b\geq (7a-8)/5$, there exists a connected graph $G$ such that $rvc(G)=a$ and $srvc(G)=b$.
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    ABSTRACT: 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 a $rainbow~tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, a tree connecting $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for every $k$-set $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$. Notice that an upper bound of the $k$-rainbow index of a graph with order $n$ is $n-1$. Chartrand et al. got that the $k$-rainbow index of a tree with order $n$ is $n-1$ and the $k$-rainbow index of a unicyclic graph with order $n$ is $n-1$ or $n-2$. Li and Sun raised an open problem to characterize the graphs of order $n$ with $rx_k(G)=n-1$ for $k\geq 3$. In an early paper we characterized the graphs of order $n$ with 3-rainbow index $n-1$. In this paper, we focus on $k=4$, and characterize the graphs of order $n$ with 4-rainbow index $n-1$.
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