Upper bound involving parameter σ 2 for the rainbow connection number

2013 ◽  
Vol 29 (4) ◽  
pp. 685-688 ◽  
Author(s):  
Jiu-ying Dong ◽  
Xue-liang Li
Keyword(s):  
Upper Bound ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

2021 ◽  
Vol 66 (3) ◽  
pp. 3-7
Author(s):  
Anh Nguyen Thi Thuy ◽  
Duyen Le Thi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Vertex Disjoint

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

scholarly journals Strong rainbow connection numbers of toroidal meshes

2018 ◽  
Vol 10 (03) ◽  
pp. 1850039
Author(s):  
Yulong Wei ◽  
Min Xu ◽  
Kaishun Wang
Keyword(s):  
Upper Bound ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Negative Answer ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

In 2011, Li et al. [The (strong) rainbow connection numbers of Cayley graphs on Abelian groups, Comput. Math. Appl. 62(11) (2011) 4082–4088] obtained an upper bound of the strong rainbow connection number of an [Formula: see text]-dimensional undirected toroidal mesh. In this paper, this bound is improved. As a result, we give a negative answer to their problem.

scholarly journals Tight Nordhaus–Gaddum-Type Upper Bound for Total-Rainbow Connection Number of Graphs

2017 ◽  
Vol 72 (4) ◽  
pp. 2079-2100 ◽  
Author(s):  
Wenjing Li ◽  
Xueliang Li ◽  
Colton Magnant ◽  
Jingshu Zhang
Keyword(s):  
Upper Bound ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

scholarly journals PENENTUAN RAINBOW CONNECTION NUMBER UNTUK AMALGAMASI GRAF LENGKAP DENGAN GRAF RODA

2019 ◽  
Vol 8 (1) ◽  
pp. 345
Author(s):  
Risya Hazani Utari ◽  
Lyra Yulianti ◽  
Syafrizal Sy
Keyword(s):  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

Suatu pewarnaan terhadap sisi-sisi di graf G terhubung tak trivial didefinisikan sebagai c : E(G) → {1, 2, · · · , k} untuk k ∈ N adalah suatu pewarnaan terhadap sisi-sisi di G sedemikian sehingga setiap sisi yang bertetangga boleh diberi warna yang sama. Banyaknya warna minimal yang diperlukan untuk membuat graf G bersifat rainbow connected disebut dengan rainbow connection number dari G, yang dinotasikan dengan rc(G). Penelitian ini menentukan rainbow connection number untuk amalgamasi 2 buah graf lengkap K4 dengan 2 buah graf roda W4 yang diperoleh dari menggabungkan satu titik pada setiap graf lengkap K4 dengan satu titik pusat pada setiap graf roda W4.Kata Kunci: Amalgamasi, Graf lengkap K4, Graf Roda W4, Rainbow Connection Number

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

2018 ◽  
Vol 7 (3) ◽  
pp. 1926 ◽  
Author(s):  
B. Praba ◽  
X.A. Benazir Obilia
Keyword(s):  
Graph Theory ◽  
Cayley Graph ◽  
Chromatic Index ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

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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