Rainbow Connection in Some Digraphs

Alva-Samos, Jesús; Montellano-Ballesteros, Juan José

doi:10.1007/s00373-016-1723-x

Cited by 14 publications

(39 citation statements)

References 12 publications

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“…In each case, we have a total-rainbow x − y geodesic in T n,k , and the claim holds. Therefore, → strc(T n,k ) ≤ k. Again (7) holds, by (1).…”

Section: Tournamentsmentioning

confidence: 78%

“…Then, there exist two elements of V ( → C n ) ∪ A( → C n ) with the same colour. We recall that for n ≥ 5, we have → rc( → C n ) = n from [1], Theorem 4; and → rvc( → C n ) = n from [9], Proposition 8. It follows that we must have a vertex and an arc with the same colour.…”

Section: Total Rainbow Connection Of Some Specific Digraphsmentioning

confidence: 97%

“…In each case, we always have a total-rainbow v i − v j geodesic, so that → strc(T N,k ) ≤ k. Therefore, (7) holds for T N,k , by (1). Finally, let n > N .…”

Section: Tournamentsmentioning

confidence: 99%

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Total rainbow connection of digraphs

Lei

Liu

Magnant

et al. 2018

Discrete Applied Mathematics

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An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of $G$ so that, any two vertices of $G$ are connected by a rainbow path (resp. rainbow geodesic). These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. Krivelevich and Yuster generalised this concept to the vertex-coloured setting. Similarly, Liu, Mestre and Sousa introduced the version which involves total-colourings. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) total rainbow connection number of digraphs. Results on the (strong) total rainbow connection number of biorientations of graphs, tournaments and cactus digraphs are presented.Comment: 29 pages, 4 figures. arXiv admin note: text overlap with arXiv:1701.0428

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“…In each case, we have a total-rainbow x − y geodesic in T n,k , and the claim holds. Therefore, → strc(T n,k ) ≤ k. Again (7) holds, by (1).…”

Section: Tournamentsmentioning

confidence: 78%

Section: Total Rainbow Connection Of Some Specific Digraphsmentioning

confidence: 97%

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Total rainbow connection of digraphs

Lei

Liu

Magnant

et al. 2018

Discrete Applied Mathematics

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“…The following statement characterizes all possible forbidden pairs X, Y for which there is a constant C XY such that rx 3 They continued to consider more and obtained an analogous result which characterizes all forbidden triples F for which there is a constant C F such that G being F -free implies rx 3 …”

Section: Problem 517 [91] For Which Families F Of Connected Graphsmentioning

confidence: 98%

“…In [3], Alva-Samos and Montellano-Ballesteros studied the rainbow connection number of cactus digraphs. A cactus digraph is a strongly connected oriented graph where every arc belongs to exactly one directed cycle.…”

Section: For Vertex-coloringmentioning

confidence: 99%

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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Rainbow Antistrong Connection in Tournaments

Wei

2020

Graphs and Combinatorics

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Rainbow Connection in Some Digraphs

Cited by 14 publications

References 12 publications

Total rainbow connection of digraphs

Total rainbow connection of digraphs

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

Rainbow Antistrong Connection in Tournaments

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