Rainbow connection numbers of Cayley graphs

Ma, Yingbin; Lu, Zaiping

doi:10.1007/s10878-016-0052-6

Cited by 18 publications

(10 citation statements)

References 10 publications

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“…resolved, see [5], [6], [11], [12], [15], [26], [29] for example. In the same way, Krivelevich and Yuster [14] introduced the concept of rainbow vertex connection number.…”

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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“…resolved, see [5], [6], [11], [12], [15], [26], [29] for example. In the same way, Krivelevich and Yuster [14] introduced the concept of rainbow vertex connection number.…”

Section: Introductionmentioning

confidence: 99%

(Strong) Proper Connection in Some Digraphs

Nie

2019

IEEE Access

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“…The diameter of Γ is the maximum distance between any two vertices in V (Γ). [1] An edge colouring of a graph G = (V, E) is a map C : E → S, where S is a set of colours, such that for all e, f ∈ E, if e and f share a vertex, then C(e) = C( f ).The chromatic index of a graph χ (G) is the minimum number of colours needed for a proper colouring of G. [2], [10], [11], [12], [13] A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path.…”

Section: Graph Theorymentioning

confidence: 99%

“…For interval graphs the rainbow connection number and diameter was obtained by Sudhakaraiah, et.al [11] in 2012.There are various research carried out on algebraic graph theory. Rainbow connection number [10], [12] of Cayley graphs was discussed by Yingbin Ma and Zaiping Lu [13] in 2017. Hengzhe li and Xueliang Li [2] have described the rainbow connection number of graphs with diameter 3.…”

Section: Introductionmentioning

confidence: 99%

Rainbow Connection Number and Chromatic Index of Rough Ideal based Rough Edge Cayley Graph

Praba¹,

Obilia²

2018

IJET

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Rainbow connection number and chromatic index are two significant parameters in the study ofgraph theory. In this work, rainbow connection number and chromatic index of Rough Ideal based Rough Edge Cayley Graph G(T(J)) are evaluated. We prove that the rainbow connection number of G(T(J)) is 2 and the chromatic index of G(T(J)) is 2(2n^m)(3m^1):Rainbow connection number and chromatic index are two significant parameters in the study of graph theory. In this work, rainbow connection number and chromatic index of Rough Ideal based Rough Edge Cayley Graph are evaluated. We prove that the rainbow connection number of is 2 and the chromatic index of is .

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“…The rainbow connection number of a graph can be used to measure the security of a communication or computer network modeled by the graph [19]. Among various types of graphs used to model networks, there are graphs of finite groups.…”

Section: Introductionmentioning

confidence: 99%

On the inverse graph of a finite group and its rainbow connection number

Umbara

Salman

Putri

2023

EJGTA

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A rainbow path in an edge-colored graph G is a path that every two edges have different colors. The minimum number of colors needed to color the edges of G such that every two distinct vertices are connected by a rainbow path is called the rainbow connection number of G. Let (Γ, * ) be a finite group with T Γ = {t ∈ Γ|t ̸ = t −1 }. The inverse graph of Γ, denoted by IG(Γ), is a graph whose vertex set is Γ and two distinct vertices, u and v, are adjacent if u * v ∈ T Γ or v * u ∈ T Γ . In this paper, we determine the necessary and sufficient conditions for the inverse graph of a finite group to be connected. We show that the inverse graph of a finite group is connected if and only if the group has a set of generators whose all elements are non-self-invertible. We also determine the rainbow connection numbers of the inverse graphs of finite groups.

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Rainbow connection numbers of Cayley graphs

Cited by 18 publications

References 10 publications

(Strong) Proper Connection in Some Digraphs

(Strong) Proper Connection in Some Digraphs

Rainbow Connection Number and Chromatic Index of Rough Ideal based Rough Edge Cayley Graph

On the inverse graph of a finite group and its rainbow connection number

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