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 Rainbow connection number of Cm o Pn and Cm o Cn

2020 ◽  
Vol 3 (2) ◽  
pp. 95
Author(s):  
Alfi Maulani ◽  
Soya Pradini ◽  
Dian Setyorini ◽  
Kiki A. Sugeng
Keyword(s):  
Connected Graph ◽  
Geodesic Path ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Different Color ◽  
Rainbow Coloring ◽  
Rainbow Color

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a nontrivial connected graph. A rainbow path is a path which is each edge colored with different color. A rainbow coloring is a coloring which any two vertices should be joined by at least one rainbow path. For two different vertices, <em>u,v</em> in <em>G</em>, a geodesic path of <em>u-v</em> is the shortest rainbow path of <em>u-v</em>. A strong rainbow coloring is a coloring which any two vertices joined by at least one rainbow geodesic. A rainbow connection number of a graph, denoted by <em>rc</em>(<em>G</em>), is the smallest number of color required for graph <em>G</em> to be said as rainbow connected. The strong rainbow color number, denoted by <em>src</em>(<em>G</em>), is the least number of color which is needed to color every geodesic path in the graph <em>G</em> to be rainbow. In this paper, we will determine  the rainbow connection and strong rainbow connection for Corona Graph <em>Cm</em> o <em>Pn</em>, and <em>Cm</em> o <em>Cn</em>.

scholarly journals Rainbow Connection Number on Amalgamation of General Prism Graph

2019 ◽  
Vol 1 (1) ◽  
Author(s):  
Rizki Hafri Yandera ◽  
Yanne Irene ◽  
Wisnu Aribowo
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Prism Graph

AbstractLet  be a nontrivial connected graph, the rainbow-k-coloring of graph G is the mapping of c: E(G)-> {1,2,3,…,k} such that any two vertices from the graph can be connected by a rainbow path (the path with all edges of different colors). The least natural number

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.

scholarly journals RAINBOW CONNECTION PADA GRAF k -CONNECTED UNTUK k = 1 ATAU 2

2013 ◽  
Vol 2 (1) ◽  
pp. 78
Author(s):  
Sally Marhelina
Keyword(s):  
Colored Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of aconnected graph G, denoted by rc(G) is the smallest number of colors needed such thatG is rainbow connected. In this paper, we will proved again that rc(G) ≤ 3(n + 1)/5 forall 3-connected graphs, and rc(G) ≤ 2n/3 for all 2-connected graphs.

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 Oriented diameter and rainbow connection number of a graph

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Xiaolong Huang ◽  
Hengzhe Li ◽  
Xueliang Li ◽  
Yuefang Sun
Keyword(s):  
Minimum Degree ◽  
Edge Coloring ◽  
Integer Number ◽  
Colored Graph ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
International Audience ◽  
Bridgeless Graph ◽  
Rainbow Path

Graph Theory International audience The oriented diameter of a bridgeless graph G is min diam(H) | H is a strang orientation of G. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and η(G), where rad(G) is the radius of G and η(G) is the smallest integer number such that every edge of G is contained in a cycle of length at most η(G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G.

scholarly journals RAINBOW CONNECTION NUMBER PADA GRAF (3K6 ∗ W6, v)

2019 ◽  
Vol 7 (3) ◽  
pp. 43
Author(s):  
Fadillah Fadillah ◽  
Lyra Yulianti ◽  
Syafrizal Sy
Keyword(s):  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Rainbow Coloring

Misalkan G = (V, E) adalah graf terhubung tak trivial. Definisikan pewarnaan c : E(G) → {1, 2, · · · , k} untuk suatu k ∈ N, dimana sisi yang bertetangga boleh diberi warna yang sama. Misalkan terdapat titik u dan v di G. Suatu lintasan-(u, v) di G dikatakan sebagai lintasan rainbow (rainbow path) jika semua sisi dalam lintasan-(u, v) tersebut memiliki warna yang berbeda. Graf G dikatakan bersifat rainbow connected terhadap pewarnaan c jika G memuat lintasan rainbow untuk setiap dua titik u dan v di G, sementara c dikatakan sebagai pewarnaan rainbow (rainbow coloring) dari G. Jika terdapat k warna yang digunakan dalam pewarnaan tersebut maka c dinamakan pewarnaan-k rainbow (rainbow k-coloring). Bilangan rainbow connection (rainbow connection number ) dari graf terhubung G, dinotasikan dengan rc(G), didefinisikan sebagai banyaknya warna minimum yang diperlukan untuk membuat graf G bersifat rainbow connected. Pada makalah ini akan ditentukan nilai bilangan rainbow connection dari graf yang merupakan hasil amalgamasi tiga graf lengkap, masing-masingnya dengan enam titik, 3K6, dengan graf roda W6, dinotasikan dengan graf (3K6 ∗ W6, v).Kata Kunci: Graf (3K6 ∗ W6, v), rainbow path, rainbow connection number

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 The rainbow connection number of 2-connected graphs

2013 ◽  
Vol 313 (19) ◽  
pp. 1884-1892 ◽  
Author(s):  
Jan Ekstein ◽  
Přemysl Holub ◽  
Tomáš Kaiser ◽  
Maria Koch ◽  
Stephan Matos Camacho ◽  
...  
Keyword(s):  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection
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