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

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.

Світобудова Михайла Гуйди

Purpose of the article is to show the particular features of the model of the universe in the genre portrait-paintings and compositions of the modern Ukrainian artist M. E. Guyda are investigated. Historico-culturalogical, comparative, iconographic and iconological methods are applied. The ideological content and the particular features of the formal solution of figurative compositions are considered in the context of the heritage of world art. The national character of the images and symbols of the house, the clan and the native land are revealed; as is the nature of the manifestation of the relationship of the earthly and the heavenly. It was found that in the center of creation in compositions of M. E. Guyda is a spiritually inspired person from the folk. The depiction of people of different ages in interaction with nature helps to convey a natural course of time. It was clarified that the European aristocratic ceremonial portrait is foundational for the Ukrainian master. Artistic-stylistic analysis revealed that in the portrait-paintings "At the Well" (2013) and "Baba Kilyna" (2016), canonical composition was transformed by the artist through the expansion of space filled with individual symbolic content. The components of the ritual ceremonial portrait (columns, draperies, table-altar) are transformed into the image of a Cossack courtyard with a hedge, a tree of life, a well, a rainbow path. In the painting "Green Festivities" (2004), the house is shown as a temple. In the composition "Chumatsky Way" (2014), the world is presented as a universe – a model of the universe-house, with the architectonics of earth and sky. Scientific novelty is that the transformation of the canonical composition of the ceremonial portrait in the work of the contemporary artist M. E. Guyda is shown. The cultural and historical content of the master’s compositions was studied in connection with the problems of national self-identification and polystylism of the art of the twentieth century.is that the appeal to archetypes and symbols allowed the artist to expand the chamber space of his native Cossack house-yard to the "Model of Guyda’s Universe". The persuasiveness of individual pictorial images and symbols is based on the unity of personal, emotional perception of life with supra-individual mythological thinking. Practical significance. The presented materials, their artistic and stylistic analysis and generalization can be used in scientific research devoted to the art of portrait painting in Ukraine and in the world.

On kernels by rainbow paths in arc-coloured digraphs

In 2018, Bai, Fujita and Zhang [Discrete Math. 341 (2018), no. 6, 1523–1533] introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph D D , which is a subset S S of vertices of D D such that ( a a ) there exists no rainbow path for any pair of distinct vertices of S S , and ( b b ) every vertex outside S S can reach S S by a rainbow path in D D . They showed that it is NP-hard to recognize whether an arc-coloured digraph has an RP-kernel and it is NP-complete to decide whether an arc-coloured tournament has an RP-kernel. In this paper, we give the sufficient conditions for the existence of an RP-kernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments, and prove that these arc-coloured digraphs have RP-kernels if certain “short” cycles and certain “small” induced subdigraphs are rainbow.

Rainbow connection number of Cm o Pn and Cm o Cn

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>.

BILANGAN TERHUBUNG TITIK PELANGI PADA AMALGAMASI GRAF BERLIAN

Suppose there is a simple, and finite graph G = (V, E). The coloring of vertices c is denoted by c: E(G) → {1,2, ..., k} with k is the number of rainbow colors on graph G. A graph is said to be rainbow connected if every pair of points x and y has a rainbow path. A path is said to be a rainbow if there are not two edges that have the same color in one path. The rainbow connected number of graph G denoted by rc(G) is the smallest positive integer-k which makes graph G has rainbow coloring. Furthermore, a graph is said to be connected to rainbow vertex if at each pair of vertices x and y there are not two vertices that have the same color in one path. The rainbow vertex connected to the number of graph G is denoted by rvc(G) is the smallest positive integer-k which makes graph G has rainbow coloring. This paper discusses rainbow vertex-connected numbers in the amalgamation of a diamond graph. A diamond graph with 2n points is denoted by an amalgamation of a diamond graph by adding the multiplication of the graph t at point v is denoted by Amal (Brn,v,t).

Rainbow Connection Number on Amalgamation of General Prism Graph

Let 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

BILANGAN RAINBOW CONNECTION UNTUK BEBERAPA GRAF CORONA SISI

Suatu lintasan uP v dikatakan sebagai rainbow path pada G jika tidak ada dua sisi pada P yang berwarna sama. Suatu graf G dikatakan rainbow-connected terhadap pewarnaan sisi-sisi, jika G memuat lintasan rainbow u − v untuk setiap dua titik u dan v pada G. Suatu pewarnaan sisi dimana G bersifat rainbow connected dinamakan rainbow coloring terhadap G. Pada tulisan ini akan ditentukan bilangan rainbow connection untuk corona sisi dari beberapa graf sederhana, yaitu rc(G H) untuk G atau H adalah graf lengkap Kn, graf lintasan Pn dan graf siklus Cn, n ≥ 3.Kata Kunci: Graf lengkap, lintasan, siklus, bilangan rainbow connection

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

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

BILANGAN RAINBOW CONNECTION UNTUK GRAF KUBIK C n;2n;n

Abstrak. Misalkan G = (V (G); E(G)) adalah suatu graf terhubung tak trivial. Suatu pewarnaanterhadap sisi-sisi di G adalah suatu pemetaan c : E(G) ! f1; 2; ; kg; k 2 N.Lintasan u v path P di G dinamakan rainbow path jika tidak terdapat dua sisi diP yang berwarna sama. Graf G disebut rainbow connected jika setiap dua titik yangberbeda di G dihubungkan oleh rainbow path. Bilangan rainbow connection dari grafterhubung G, ditulis rc(G), didenisikan sebagai banyaknya warna minimal yang diperlukanuntuk membuat graf G bersifat rainbow connected. Suatu Graf Kubik Cmerupakan graf kubik terhubung yang terbentuk dari tiga lingkaran dengan banyak titikpada lingkaran pertama sama dengan lingkaran ketiga yaitu sebanyak n dan banyak titikpada lingkaran kedua sebanyak 2n dengan himpunan sisi Emerupakan himpunan sisiyang menghubungkan lingkaran ke-i dengan lingkaran ke- i + 1. Pada paper ini ditunjukkanbahwa rc(C5;10;5) = 7 dan rc(C6;12;6) = 8.Kata Kunci: Graf kubik, graf lingkaran, bilangan rainbow connectionin;2n;n