Rainbow Connections of Graphs: A Survey

Li, Xueliang; Shi, Yongtang; Sun, Yuefang

doi:10.1007/s00373-012-1243-2

Cited by 241 publications

(156 citation statements)

References 70 publications

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“…There is a survey [104] and a book [112] which have been published on this topic. More and more researchers are working in this field, and many new papers have been published in journals.…”

Section: By Definition If H Is a Connected Spanning Subgraph Of G Tmentioning

confidence: 99%

“…In particular, the following problem could be interesting but may be difficult: Problem 2.4. [104] Characterize those graphs G with rc(G) = diam(G), or give some sufficient conditions to guarantee that rc(G) = diam(G). Similar problems for the parameter src(G) can be proposed.…”

Section: Rainbow Connection Coloring Of Edge-versionmentioning

confidence: 99%

“…[104] Determine the relationship between rc(G) and rc(L(G)), is there an upper bound for one of these parameters in terms of the other?…”

Section: Graph Classesmentioning

confidence: 99%

See 2 more Smart Citations

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

Li¹,

Sun²

2017

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The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow k-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study.

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“…There is a survey [104] and a book [112] which have been published on this topic. More and more researchers are working in this field, and many new papers have been published in journals.…”

Section: By Definition If H Is a Connected Spanning Subgraph Of G Tmentioning

confidence: 99%

Section: Rainbow Connection Coloring Of Edge-versionmentioning

confidence: 99%

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An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

Li¹,

Sun²

2017

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“…Over the past years, this strengthened notion of connectivity has received a significant amount of attention in the research community. The applications of rainbow connectivity are discussed in detail for instance in the recent survey [24], and various bounds are also available in [10,25].…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Rainbow Coloring Problems

Eiben

Ganian

Lauri

2016

Lecture Notes in Computer Science

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Abstract. An edge-colored graph G is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by rc(G), is the minimum number of colors needed to make G rainbow connected. Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph-theoretic points of view. In this paper we present a range of new results on the computational complexity of computing the four major variants of the rainbow connection number. In particular, we prove that the Strong Rainbow Vertex Coloring problem is NP-complete even on graphs of diameter 3. We show that when the number of colors is fixed, then all of the considered problems can be solved in linear time on graphs of bounded treewidth. Moreover, we provide a linear-time algorithm which decides whether it is possible to obtain a rainbow coloring by saving a fixed number of colors from a trivial upper bound. Finally, we give a linear-time algorithm for computing the exact rainbow connection numbers for three variants of the problem on graphs of bounded vertex cover number.

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“…Other basic facts established in [7] are that rc(G) = 1 if and only if G is a clique and rc(G) = |V (G)| − 1 if and only if G is a tree. Besides its theoretical interest, rainbow connectivity is also of interest in applied settings, such as securing sensitive information [13], transfer and networking [5].…”

Section: Introductionmentioning

confidence: 99%