Rainbow Connections of Graphs: A Survey

Graphs and Combinatorics (Impact Factor: 0.39). 01/2013; 29(1):1--38. DOI: 10.1007/s00373-012-1243-2
Source: arXiv


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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Available from: Xueliang Li, Aug 21, 2014
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    • "The minimum number of colors required to rainbow color a graph G is called the rainbow connection number, denoted by rc(G). For more results on the rainbow connection, we refer to the survey paper [21] of Li, Shi and Sun and a new book [22] of Li and Sun. "
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    ABSTRACT: The concept of monochromatic connectivity was introduced by Caro and Yuster. A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} ($MC$-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an $MC$-coloring of a graph $G$. In this paper, we study the monochromatic connection number on the lexicographical, strong, Cartesian and direct product and present several upper and lower bounds for these products of graphs.
    Discrete Mathematics Algorithms and Applications 05/2015; DOI:10.1142/S1793830916500117
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    • "See for example, Caro et al. [2], Chartrand et al. [6], and Krivelevich and Yuster [12]. Recently, Li et al. [14], and Li and Sun [17], published a survey and a book on the current status of rainbow connection. "
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    ABSTRACT: An edge-coloured path is rainbow if the colours of its edges are distinct. For a positive integer k, an edge-colouring of a graph G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The rainbow k-connection number rc k (G) is defined to be the minimum integer t such that there exists an edge-colouring of G with t colours which is rainbow k-connected. We consider rc2(G) when G has fixed vertex-connectivity. We also consider rc k (G) for large complete bipartite and multipartite graphs G with equipartitions. Finally, we determine sharp threshold functions for the properties rc k (G) = 2 and rc k (G) = 3, where G is a random graph. Related open problems are posed.
    Journal of Combinatorial Mathematics and Combinatorial Computing 05/2015; 93.
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    • "Chartrand, Johns, McKeon and Zhang introduced in [3] the concept of rainbow connection in graphs, and since then such topic has been broadly studied, see [5] for a survey on the matter. The rainbow connection for digraphs was first presented by Dorbec, Schiermeyer, Sidorowicz and Sopena in [4]. "
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    ABSTRACT: An arc-coloured digraph $D$ is said to be \emph{rainbow connected} if for every two vertices $u$ and $v$ there is an $uv$-path all whose arcs have different colours. The minimun number of colours required to make the digraph rainbow connected is called the \emph{rainbow connection number} of $D$, denoted $\stackrel{\rightarrow}{rc}(D)$. In \cite{Dorbec} it was showed that if $T$ is a strong tournament with $n\geq 5$ vertices, then $2\leq \stackrel{\rightarrow}{rc}(T)\leq n-1$; and that for every $n$ and $k$ such that $3\leq k\leq n-1$, there exists a tournament $T$ on $n$ vertices such that $\stackrel{\rightarrow}{rc}(T)=k$. In this note it is showed that for any $n\ge6$, there is a tournament $T$ of $n$ vertices such that $\stackrel{\rightarrow}{rc}(T)=2$.
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