scholarly journals A Note on the Rainbow Connection of Random Regular Graphs

10.37236/6789 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Michael Molloy
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Connection Number ◽  
Open Case ◽  
Rainbow Connection Number ◽  
Rainbow Connection

We prove that a random 3-regular graph has rainbow connection number $O(\log n)$. This completes the remaining open case from Rainbow connection of random regular graphs, by Dudek, Frieze and Tsourakakis.

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 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 .

scholarly journals The Rainbow Connection Number of Triangular Snake Graphs

2020 ◽  
Author(s):  
Aleffer Rocha ◽  
Sheila M. Almeida ◽  
Leandro M. Zatesko
Keyword(s):  
Information Security ◽  
Edge Coloring ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Proper Edge Coloring ◽  
Rainbow Connection ◽  
Rainbow Coloring

Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention last years in Combinatorics. The rainbow connection number of a graph G is the least number of colors for a (not necessarily proper) edge coloring of G such that between any pair of vertices there is a path whose edge colors are all distinct. In this paper we determine the rainbow connection number of the triple triangular snake graphs.

scholarly journals Graphs with rainbow connection number two

2011 ◽  
Vol 31 (2) ◽  
pp. 313 ◽  
Author(s):  
Arnfried Kemnitz ◽  
Ingo Schiermeyer
Keyword(s):  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

scholarly journals RAINBOW CONNECTION PADA BEBERAPA GRAF

2013 ◽  
Vol 2 (2) ◽  
pp. 17
Author(s):  
Gema Hista Medika
Keyword(s):  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Rainbow Coloring

Misalkan G adalah graf terhubung tak-trivial. Denisikan pewarnaan c :E(G) ! f1; 2; :::; kg, k 2 N, dimana dua sisi yang bertetangga boleh memiliki warnayang sama. Suatu u 􀀀 v path P di G dikatakan rainbow path jika tidak ada dua sisi diP yang memiliki warna sama. Graf G dikatakan rainbow connected jika setiap dua titikyang berbeda di G dihubungkan oleh rainbow path. Pewarnaan sisi yang menyebabkan Gbersifat rainbow connected dikatakan rainbow coloring. Rainbow connection number darigraf terhubung G, ditulis rc(G), didenisikan sebagai banyaknya warna minimal yangdiperlukan untuk membuat graf G bersifat rainbow connected. Misalkan c adalah rainbowcoloring dari graf terhubung G. Untuk dua titik u dan v di G, rainbow u-v geodesic padaG adalah rainbow u-v path yang panjangnya d(u; v), dimana d(u; v) adalah jarak antarau dan v (panjang u-v path terpendek di G). Graf G dikatakan strongly rainbow-connectedjika G memiliki suatu rainbow u-v geodesic untuk setiap dua titik u dan v di G. Mini-mum k yang terdapat pada pewarnaan c : E(G) ! f1; 2; :::; kg sedemikian sehingga Gadalah strongly rainbow-connected dikatakan strong rainbow connection number, src(G);di G. Jadi, rc(G) src(G) untuk setiap graf terhubung di G. Pada paper ini akan di-ulas kembali tentang strong rainbow connection number dari graf bipartit lengkap Ks;tdengan 1 s t dimana s; t 2 N adalah src(Ks;t) = d spte, sedangkan rainbow connec-tion number dari graf bipartit lengkap Ks;t dengan 2 s t dimana s; t 2 N adalahrc(Ks;t) = minfd spte; 4g.

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