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

Li, Xueliang; Sun, Yuefang

doi:10.20429/tag.2017.000103

Cited by 46 publications

(35 citation statements)

References 81 publications

(158 reference statements)

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“…For more results, the reader can see [7], [13], [22]- [24] for details. The survey [18] and a new monograph [17] have given more results on the rainbow connectivity of graphs.…”

Section: Introductionmentioning

confidence: 99%

(Strong) Proper Connection in Some Digraphs

Nie

2019

IEEE Access

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An arc-colored digraph D is proper connected if any pair of vertices v i , v j ∈ V (D) there is a proper v i − v j path whose adjacent arcs have different colors and a proper v j − v i path whose adjacent arcs have different colors. The proper connection number of a digraph D is the minimum number of colors needed to make D proper connected, denoted by − → pc(D). An arc-colored digraph D is strong proper connected if any pair of vertices v i , v j ∈ V (D) there is a proper v i − v j geodesic and a proper v j − v i geodesic. The strong proper connection number of D is the minimum number of colors required to color the arcs of D in order to make D strong proper connected, denoted by − → spc(D). In this paper, we will show some results on − → pc(D) and − → spc(D), mostly for the case of the (strong) proper connection numbers of cacti and circulant digraphs. INDEX TERMS Proper path, proper connection number, proper geodesic, strong proper connection number.

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“…For more results, the reader can see [7], [13], [22]- [24] for details. The survey [18] and a new monograph [17] have given more results on the rainbow connectivity of graphs.…”

Section: Introductionmentioning

confidence: 99%

(Strong) Proper Connection in Some Digraphs

Nie

2019

IEEE Access

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“…These concepts were first introduced by Chartrand et al in [2] and have been well-studied since then. For further details, we refer the reader to a survey [4] (with an updated version available at [5]) and a book [6].…”

Section: Introductionmentioning

confidence: 99%

On proper (strong) rainbow connection of graphs

Jiang¹,

Li²,

Li³

et al. 2021

Discuss. Math. Graph Theory

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A path in an edge-colored graph G is called a rainbow path if no two edges on the path have the same color. The graph G is called rainbow connected if between every pair of distinct vertices of G, there is a rainbow path. Recently, Johnson et al. considered this concept with the additional requirement that the coloring of G is proper. The proper rainbow connection number of G, denoted by prc(G), is the minimum number of colors needed to properly color the edges of G so that G is rainbow connected. Similarly, the proper strong rainbow connection number of G, denoted by psrc(G), is the minimum number of colors needed to properly color the edges of G such that for any two distinct vertices of G, there is a rainbow geodesic (shortest path) connecting them. In this paper, we characterize those graphs with proper rainbow connection numbers equal to the size or within 1 of the 2 H. Jiang, W. Li, X. Li and C. Magnant size. Moreover, we completely solve a question proposed by Johnson et al. by proving that if G = K p1 • • • K pn , where n ≥ 1, and p 1 ,. .. , p n > 1 are integers, then prc(G) = psrc(G) = χ ′ (G), where χ ′ (G) denotes the chromatic index of G. Finally, we investigate some sufficient conditions for a graph G to satisfy prc(G) = rc(G), and make some slightly positive progress by using a relation between rc(G) and the girth of the graph.

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“…Meanwhile, some results about rainbow connection number of graphs resulted from graph operations were also investigated by some researchers, such as Cartesian product [16,1], strong product [1,10], lexicographic product [16,1,10], direct product [10], corona product [6], and amalgamation [8]. Interested readers can see [17,19] for details on this topic.…”

Section: Introductionmentioning

confidence: 99%

The rainbow 2-connectivity of Cartesian products of 2-connected graphs and paths

Susanti

Salman

Simanjuntak

2020

ejgta

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An edge-colored graph G is rainbow k-connected, if there are k-internally disjoint rainbow paths connecting every pair of vertices of G. The rainbow k-connection number of G, denoted by rc k (G), is the minimum number of colors needed for which there exists a rainbow k-connected coloring for G. In this paper, we are able to find sharp lower and upper bounds for the rainbow 2-connection number of Cartesian products of arbitrary 2-connected graphs and paths. We also determine the rainbow 2-connection number of the Cartesian products of some graphs, i.e. complete graphs, fans, wheels, and cycles, with paths.

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

Cited by 46 publications

References 81 publications

(Strong) Proper Connection in Some Digraphs

(Strong) Proper Connection in Some Digraphs

On proper (strong) rainbow connection of graphs

The rainbow 2-connectivity of Cartesian products of 2-connected graphs and paths

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