Rainbow Connections of Graphs: A Survey

Graphs and Combinatorics (Impact Factor: 0.33). 01/2013; 29(1):1--38. DOI: 10.1007/s00373-012-1243-2
Source: arXiv

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: The rainbow connection problem belongs to the class of NP-Hard graph theoretical problems. 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. In this study, we present a new mathematical model for the rainbow connection problem.
    2013 7th International Conference on Application of Information and Communication Technologies (AICT); 10/2013
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    ABSTRACT: The proper connection number $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one path in $G$ such that no two adjacent edges of the path are colored the same, and such a path is called a proper path. In this paper, we show that for every connected graph with diameter 2 and minimum degree at least 2, its proper connection number is 2. Then, we give an upper bound $\frac{3n}{\delta + 1}-1$ for every connected graph of order $n$ and minimum degree $\delta$. We also show that for every connected graph $G$ with minimum degree at least $2$, the proper connection number $pc(G)$ is upper bounded by $pc(G[D])+2$, where $D$ is a connected two-way (two-step) dominating set of $G$. Bounds of the form $pc(G)\leq 4$ or $pc(G)=2$, for many special graph classes follow as easy corollaries from this result, which include connected interval graphs, asteroidal triple-free graphs, circular arc graphs, threshold graphs and chain graphs, all with minimum degree at least $2$. Furthermore, we get the sharp upper bound 3 for the proper connection numbers of interval graphs and circular arc graphs through analyzing their structures.
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    ABSTRACT: 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 paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a \emph{vertex-rainbow $S$-tree} if the vertices of $V(T)\setminus S$ have distinct colors. For a fixed integer $k$ with $2\leq k\leq n$, the vertex-coloring $c$ of $G$ is called a \emph{$k$-vertex-rainbow coloring} if for every $k$-subset $S$ of $V(G)$ there exists a vertex-rainbow $S$-tree. In this case, $G$ is called \emph{vertex-rainbow $k$-tree-connected}. The minimum number of colors that are needed in a $k$-vertex-rainbow coloring of $G$ is called the \emph{$k$-vertex-rainbow index} of $G$, denoted by $rvx_k(G)$. When $k=2$, $rvx_2(G)$ is nothing new but the vertex-rainbow connection number $rvc(G)$ of $G$. In this paper, sharp upper and lower bounds of $srvx_k(G)$ are given for a connected graph $G$ of order $n$,\ that is, $0\leq srvx_k(G)\leq n-2$. We obtain the Nordhaus-Guddum results for $3$-vertex-rainbow index, and show that $rvx_3(G)+rvx_3(\overline{G})=4$ for $n=4$ and $2\leq rvx_3(G)+rvx_3(\overline{G})\leq n-1$ for $n\geq 5$. Let $t(n,k,\ell)$ denote the minimal size of a connected graph $G$ of order $n$ with $rvx_k(G)\leq \ell$, where $2\leq \ell\leq n-2$ and $2\leq k\leq n$. The upper and lower bounds for $t(n,k,\ell)$ are also obtained.

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