On the Complexity of Rainbow Coloring Problems

Eiben, Eduard; Ganian, Robert; Lauri, Juho

doi:10.1007/978-3-319-29516-9_18

Cited by 7 publications

(10 citation statements)

References 44 publications

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“…The problem of Strong Rainbow Vertex k-Connection Coloring (k-SRVC) are then defined analogously for srvc(G). In [48], the authors considered the hardness of the problem k-SRVC.…”

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

“…[48] There is no polynomial time algorithm for approximating the strong rainbow vertex connection number of an n-vertex graph of bounded diameter within a factor of n 1/2−ǫ for any ǫ, unless P = NP .…”

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

“…[48] Let p ∈ N be fixed. Then the problems k-RVC and k-SRVC can be solved in time O(n) on n-vertex graphs of treewidth at most p. Furthermore, k-RVC, k-SRVC can be solved in time O(n 3 ) on n-vertex graphs of clique-width at most p.…”

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

“…In [48], the authors also considered the "saving" versions of the problem, which ask whether it is possible to improve upon the trivial upper bound for the number of colors. The problem of Saving k Rainbow Vertex Colors (k-SavingRVC) is stated as follows: for a connected undirected graph G, does rvc(G) ≤ |E(G)| − k hold?…”

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

“…[48] Let p ∈ N be fixed. Then the problems RVC and SRVC can be solved in time O(n) on n-vertex graphs of vertex cover number at most p.…”

Section: Rainbow Connection Coloring Of Vertex-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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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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“…The problem of Strong Rainbow Vertex k-Connection Coloring (k-SRVC) are then defined analogously for srvc(G). In [48], the authors considered the hardness of the problem k-SRVC.…”

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

Section: Rainbow Connection Coloring Of Vertex-versionmentioning

confidence: 99%

“…[48] Let p ∈ N be fixed. Then the problems RVC and SRVC can be solved in time O(n) on n-vertex graphs of vertex cover number at most p.…”

Section: Rainbow Connection Coloring Of Vertex-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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Algorithms and Bounds for Very Strong Rainbow Coloring

Chandran

Das

Issac

et al. 2018

Lecture Notes in Computer Science

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A well-studied coloring problem is to assign colors to the edges of a graph G so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (src(G)) of the graph. When proving upper bounds on src(G), it is natural to prove that a coloring exists where, for every shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call very strong rainbow connection number (vsrc(G)). In this paper, we give upper bounds on vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on src(G) for these classes, showing that the study of vsrc(G) enables meaningful progress on bounding src(G). Then we study the complexity of the problem to compute vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for src(G). We also observe that deciding whether vsrc(G) = k is fixed-parameter tractable in k and the treewidth of G. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether vsrc(G) ≤ 3 nor to approximate vsrc(G) within a factor n 1−ε , unless P=NP.

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On the complexity of rainbow coloring problems

Eiben

Ganian

Lauri

2018

Discrete Applied Mathematics

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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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On the Complexity of Rainbow Coloring Problems

Cited by 7 publications

References 44 publications

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

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

Algorithms and Bounds for Very Strong Rainbow Coloring

On the complexity of rainbow coloring problems

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