The rainbow connection number of some subdivided roof graphs

Susanti, Bety Hayat; Salman, A. N. M.; Simanjuntak, Rinovia

doi:10.1063/1.4940822

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…There are also some results about bounds for rainbow connection number of graphs resulted from graph operations; for instance: Cartesian product graphs [12,15], composition (lexicographic product) graphs [10,15], join of graphs [15], direct product and strong product graphs [10], and amalgamation of some graphs [9]. Some other results on rainbow connection number of graphs can be found in [11,16,17,21,22]. An overview about rainbow connection number can be found in a survey by Li et al [13] and a book of Li and Sun [14].…”

Section: Introductionmentioning

confidence: 99%

Rainbow connection number of comb product of graphs

Fitriani¹,

Salman

Awanis

2022

EJGTA

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An edge-colored graph G is called a rainbow connected if any two vertices are connected by a path whose edges have distinct colors. Such a path is called a rainbow path. The smallest number of colors required in order to make G rainbow connected is called the rainbow connection number of G. For two connected graphs G and H with v ∈ V (H), the comb product between G and H, denoted by G ▷ v H, is a graph obtained by taking one copy of G and |V (G)| copies of H and identifying the i-th copy of H at the vertex v to the i-th vertex of G. In this paper, we give sharp lower and upper bounds for the rainbow connection number of comb product between two connected graphs. We also determine the exact values of rainbow connection number of G ▷ v H for some connected graphs G and H.

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Section: Introductionmentioning

confidence: 99%

Rainbow connection number of comb product of graphs

Fitriani¹,

Salman

Awanis

2022

EJGTA

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“…Numerous authors have investigated bounds, algorithms, and computational complexity of the rainbow connection number of some graphs. Some results about rainbow connection number of certain graphs have been determined by some researchers, such as, complete graphs, trees, complete bipartite graphs, and complete multipartite graphs [3], rocket graphs [21], pencil graphs [24], flower graphs [14], origami graphs and pizza graphs [27], stellar graphs [23], and some subdivided roof graphs [26]. In 2018, Septyanto and Sugeng [22] generalized the notion of "color codes" that was originally used by Chartrand et al [3] in their study of the rc of complete bipartite graphs, so that it can be applied to any connected graph.…”

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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“…Therefore, many previous researchers studied the 𝑟𝑐 of graphs by limiting their study to certain classes of graphs, e.g. [2], [5], [6], [7], [8], [9], [10], [11]. We refer the readers to [12], [13] for some detailed surveys on 𝑟𝑐 of graphs.…”

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confidence: 99%

The Strong 3-Rainbow Index of Graphs Containing Three Cycles

Awanis

2023

InPrime:Ind.Jour.Pure.Applied.Math

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The concept of a strong k-rainbow index is a generalization of a strong rainbow connection number, which has an interesting application in security systems in a communication network. Let G be an edge-colored connected graph of order n, where adjacent edges may be colored the same. A rainbow tree in G is a tree whose edges have distinct colors. For an integer k with 2≤k≤n, the strong k-rainbow index srx_k (G) of G is the minimum number of colors needed to color all edges of G so that every k vertices of G are connected by a rainbow tree of minimum size. We focus on k=3. It is clear that srx_3 (G)≤‖G‖, where the upper bound is sharp since the srx_3 of a tree equals its size. Hence, we are interested in studying how the srx_3 of a tree changes if we add some edges connecting two nonadjacent vertices in the tree. This paper is focused on graphs containing three cycles. We first determine a sharp upper bound of the srx_3 of graphs containing exactly three edge-disjoint cycles. We also determine the exact values of srx_3 of theta graph θ(a_1,a_2,a_3) for certain values of a_1, a_2, and a_3.Keywords: cycle; rainbow coloring; rainbow Steiner tree; theta graph; tree. AbstrakKonsep indeks pelangi-k kuat merupakan perumuman dari bilangan terhubung pelangi kuat yang memiliki aplikasi menarik dalam sistem keamanan jaringan komunikasi. Misalkan G adalah suatu graf terhubung berorde n yang memiliki suatu pewarnaan sisi, dimana dua sisi bertetangga boleh memiliki warna yang sama. Pohon pelangi di G adalah pohon yang setiap sisinya memiliki warna berbeda. Untuk suatu bilangan bulat k dengan 2≤k≤n, indeks pelangi-k kuat srx_k (G) graf G adalah banyak warna minimum yang dibutuhkan untuk mewarnai semua sisi di G sehingga setiap k titik di G dihubungkan oleh suatu pohon pelangi berukuran minimum. Kami fokus pada k=3. Jelas bahwa srx_3 (G)≤‖G‖, dimana batas atas ini merupakan batas ketat karena srx_3 pohon sama dengan ukurannya. Karena itu, kami tertarik untuk mempelajari bagaimana srx_3 pohon berubah jika ditambahkan beberapa sisi yang menghubungkan dua titik tidak bertetangga di pohon tersebut. Artikel ini difokuskan pada graf yang memuat tiga siklus. Pertama, kami menentukan batas atas ketat srx_3 graf yang memuat tepat tiga siklus saling lepas sisi. Kami juga menentukan nilai eksak srx_3 graf theta θ(a_1,a_2,a_3 ) untuk beberapa nilai a_1, a_2, dan a_3 tertentu.Kata Kunci: siklus; pewarnaan pelangi; pohon Steiner pelangi; graf theta; pohon. 2020MSC: 05C05, 05C15, 05C38, 05C40.

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The rainbow connection number of some subdivided roof graphs

Cited by 4 publications

References 3 publications

Rainbow connection number of comb product of graphs

Rainbow connection number of comb product of graphs

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

The Strong 3-Rainbow Index of Graphs Containing Three Cycles

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