Article

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: 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: Let k be a positive integer and G be a k-connected graph. An edge-coloured path is rainbow if its edges have distinct colours. The rainbow k-connection number of G, denoted by rc k (G), is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function rc k (G) was first introduced by G. Chartrand et al. [Math. Bohem. 133, No. 1, 85–98 (2008; Zbl 1199.05106)], and has since attracted considerable interest. In this paper, we consider a version of the function rc k (G) which involves vertex-colourings. A vertex-coloured path is vertex-rainbow if its internal vertices have distinct colours. The rainbow vertex k-connection number of G, denoted by rvc k (G), is the minimum number of colours required to colour the vertices of G so that any two vertices of G are connected by k internally vertex-disjoint vertex-rainbow paths. We shall study the function rvc k (G) when G is a cycle, a wheel, and a complete multipartite graph. We also construct graphs G where rc k (G) is much larger than rvc k (G) and vice versa so that we cannot in general bound one of rc k (G) and rvc k (G) in terms of the other.
    Discrete Applied Mathematics 11/2013; 161(16). · 0.68 Impact Factor
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    Discrete Applied Mathematics 01/2014; · 0.68 Impact Factor
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    ABSTRACT: A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the minimum number of colors that are needed to color the edges of $G$ such that there exists a rainbow path connecting every two vertices of $G$. Similarly, a tree in $G$ is a rainbow~tree if no two edges of it receive the same color. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow tree connecting $S$ for each $k$-subset $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$, where $k$ is an integer such that $2\leq k\leq n$. In \cite{Chakraborty}, the authors got the following result: For every $\epsilon> 0$, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connection, where the bound depends only on $\epsilon$. In \cite{Krivelevich and Yuster}, the authors proved that if $G$ has $n$ vertices and the minimum degree $\delta(G)$ then $rc(G)<20n/\delta(G)$. This bound was later improved to $3n/(\delta(G)+1)+3$ in \cite{Chandran}. Since $rc(G)=rx_2(G)$, a natural problem arises: for a general $k$ determining the true behavior of $rx_k(G)$ as a function of the minimum degree $\delta(G)$. In this paper, we give upper bounds of $rx_k(G)$ in terms of the minimum degree $\delta(G)$ in different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected $2$-step dominating sets, connected $(k-1)$-dominating sets and $k$-dominating sets of $G$.
    07/2014;

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