Rainbow Vertex-connection Number Research Articles

Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Graphs with Large Clique Number

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] to make [Formula: see text] rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] to make [Formula: see text] strong rainbow vertex connected. These two concepts are inspired by the concepts of proper (strong) rainbow connection number of graphs. In this paper, we determine the values of [Formula: see text] and [Formula: see text] of [Formula: see text] with large clique numbers [Formula: see text] and [Formula: see text]. Moreover, we determine the values of [Formula: see text] and [Formula: see text] of [Formula: see text] with large clique numbers [Formula: see text], [Formula: see text] and [Formula: see text].

BILANGAN KETERHUBUNGAN TITIK PELANGI BEBERAPA KELAS GRAF

A graph G is called a rainbow vertex connected if every two vertices G are connected by a rainbow path, that is, a path whose all the internal vertices are of a different color. The rainbow vertex connection number of graph G denoted by rvc(G) is the minimum number of colors used to color all vertices by G such that the graph G is connected to rainbow vertex. The rainbow vertex connection number in a graph will not be less than the diameter of the graph minus one. The rainbow vertex connection number discussed in this article for various classes of graphs include complete graph Kn, complete bipartite graph Km,n , wheel graph Wn , two-layer wheel graph Wn2, complete multipartite graph Kn1,n2,...,nt , path Pn, comb graph GSn, graph , graph , graph , graph .
 Keywords: graph, vertex coloring, rainbow vertex connection number.

The rainbow vertex connection number of ladder graphs and Roach graphs

A vertex-coloured graph G is said to be rainbow vertex-connected, if every two vertices of G are connected by a path whose internal vertices have distinct colours. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colours that are needed to make G, a rainbow vertex-connected. This study focuses on deriving formulas for the rainbow vertex connectivity number of a simple ladder graph and a roach graph.

RAINBOW VERTEX-CONNECTION NUMBER ON COMB PRODUCT OPERATION OF CYCLE GRAPH (C_4) AND COMPLETE BIPARTITE GRAPH (K_(3,N))

Rainbow vertex-connection number is the minimum colors assignment to the vertices of the graph, such that each vertex is connected by a path whose edges have distinct colors and is denoted by . The rainbow vertex connection number can be applied to graphs resulting from operations. One of the methods to create a new graph is to perform operations between two graphs. Thus, this research uses comb product operation to determine rainbow-vertex connection number resulting from comb product operation of cycle graph and complete bipartite graph & . The research finding obtains the theorem of rainbow vertex-connection number at the graph of for while the theorem of rainbow vertex-connection number at the graph of for for .

Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Some Special Graphs

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is strong rainbow vertex connected. These two concepts are inspired by the concept of proper (strong) rainbow connection number of graphs. In this paper, we first determine the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], pencil graphs, circular ladders or Möbius ladders. Secondly, we obtain the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], paths, cycles, wheels, complete multipartite graphs, pencil graphs, circular ladders and Möbius ladders. Finally, we characterize all the connected graphs [Formula: see text] with [Formula: see text] and [Formula: see text].

Rainbow Vertex Antimagic Coloring 2-Connection paada Keluarga Graf Tangga

All graph in this paper are connected graph and simple graph. Let G = (V,E)be a connected graph. Rainbow vertex connection is the assignment of G that has interior vertices with different colors. The minimum number of colors from the rainbow vertex coloring in graph G is called rainbow vertex connection number. If wf(u) ̸= wf(v) for two different vertext u, v ∈ V (G) then f is called antimagic labeling for graph G. Rainbow vertex antimagic coloring is a combination between rainbow coloring and antimagic labeling. Graph G is called rainbow vertex antimagic coloring 2-connection if G has at least 2 rainbow paths from u − v. Rainbow vertex antimagic coloring 2-connection to denoted as rvac2(G). In this paper, we will study rainbow vertex antimagic coloring 2-connection on a family of graphs ladder that includes H-graph Hn for n ≥ 2, slide ladder graph SLn for n ≥ 2, and graph Octa-Chain OCn for n ≥ 2.

The Rainbow-Vertex Connection Number [RVCN] of Subdivision of Certain Graphs

Rainbow-Vertex Connection Number [rvcn] is computed for some graphs by the researchers. Here we have considered the subdivision graphs of certain graph classes. The rainbow edge connection number of subdivision of Triangular snake graph was already found[1]. Using the definition of rainbow-vertex connection number [5], which is the smallest positive integer k such that the graph is rainbow-vertex connected, we find the rainbow vertex number of subdivision graph of Friendship graph ,, Triangular snake graph and Comb graph .

Rainbow Vertex Connection Number pada Keluarga Graf Roda

The rainbow vertex connection was first introduced by krivelevich and yuster in 2009 which is an extension of the rainbow connection. Let graph $G =(V,E)$ is a connected graph. Rainbow vertex-connection is the assignment of color to the vertices of a graph $G$, if every vertex on graph $G$ is connected by a path that has interior vertices with different colors. The minimum number of colors from the rainbow vertex coloring in graph $G$ is called rainbow vertex connection number which is denoted $rvc(G)$. The result of the research are the rainbow vertex connection number of family wheel graphs.

Analisis rainbow vertex connection pada beberapa graf khusus dan operasinya

The vertex colored graph G is said rainbow vertex cennected, if for every two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. On this research, will be raised the issue of how to produce graphs the results of some special graph and how to find the rainbow vertex connection. Operation that use cartesian product, crown product, and shackle. Theorem in this research rainbow vertex connection number in graph the results of operations Wd3,m □ Pn,,Wd3,m ⵙ Pn, and shack(Btm,v,n).

Rainbow vertex-connection number on a small-world Farey graph

A small-world Farey graph is a recursively constructed graph characterized as a small-world network graph. Many of the network properties of this graph has been studied. In this paper, we take a combinatorial approach to investigate the rainbow vertex connectivity of the small-world Farey graph. A graph is rainbow vertex-connected when there exists a path, in which all of its internal vertices have distinct colors, between each pair of vertices. We give a coloring so that the graph is rainbow vertex-connected. The rainbow vertex-connection number of the small-world Farey graph, which is the minimum number of colors needed so that the graph is rainbow vertex-connected, is one less than the diameter of the graph.

Bilangan Terhubung Titik Pelangi pada Graf Hasil Operasi Korona Graf Prisma (P_(m,2)) dan Graf Lintasan (P_3)

Rainbow vertex-connection number is the minimum k-coloring on the vertex graph G and is denoted by rvc(G). Besides, the rainbow-vertex connection number can be applied to some special graphs, such as prism graph and path graph. Graph operation is a method used to create a new graph by combining two graphs. Therefore, this research uses corona product operation to form rainbow-vertex connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2). The results of this study obtain that the theorem of rainbow vertex-connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2) for 3 = m = 7 are rvc (G) = 2m rvc (G) = 2.

On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion

All graph in this paper are simple, finite, and connected. Let be a labeling of a graph . The function is called antimagic rainbow edge labeling if for any two vertices and , all internal vertices in path have different weight, where the weight of vertex is the sum of its incident edges label. The vertex weight denoted by for every . If G has a antimagic rainbow edge labeling, then is a antimagic rainbow vertex connection, where the every vertex is assigned with the color . The antimagic rainbow vertex connection number of , denoted by , is the minimum colors taken over all rainbow vertex connection induced by antimagic rainbow edge labeling of . In this paper, we determined the exact value of the antimagic rainbow vertex connection number of path ( ), wheel ( ), friendship ( ), and fan ( ).

Analisis Rainbow Vertex Connection pada Beberapa Graf Khusus dan Operasinya

Suppose $G=(V(G),E(G))$ is a non-trivial connected graph with edge coloring defined as $c:E(G) \rightarrow \{1,2,...,k\} ,k \in N$, with the condition that neighboring edges can be the same color. An original path is {\it rainbow path} if there are no two edges in the path of the same color. The graph $G$ is called rainbow connected if every two vertices in $G$ with rainbow path in $G$. The coloring here is called rainbow coloring, and the minimal coloring in a graph $G$ rainbow connection number is denoted by $rc(G)$. Suppose $G=(V(G),E(G))$ is a non-trivial connected graph with a vertex coloring defined as $c':V(G) \rightarrow \{1,2,...,k\},k \in N$, with the condition that neighboring interior vertex may have the same color. An original path is rainbow vertex path if there are no two vertices in the path of the same color. The graph $G$ is called rainbow vertex connected if every two vertices in $G$ with rainbow vertex path in $G$. The $G$ coloring is called rainbow vertex coloring, and the minimal coloring in a $G$ graph is called rainbow vertex connection number which is denoted by $rvc(G)$. This research produces rainbow vertex connection number on the graph resulting from the operation \emph{amal}($Bt_{m}$, $v$, $n$), $Wd_{3,m}$ $\Box$ $ P_n$, $P_m$ $\odot$ $Wd_{3,n}$, $Wd_{3,m}$ $+$ $C_n$, and \emph{shack}($Bt_{m}$, $v $, $n$).

On The Locating Rainbow Connection Number of A Graph

Let k be a positive integer and G = (V(G), E(G)) be a finite and connected graph. A rainbow vertex k-coloring of G is a function c: V(G) → {1,2,…, k} such that for every two vertices u and v in V(G) there exists a u-v path whose internal vertices have distinct colors. Such path is called a rainbow vertex path. The rainbow vertex connection number of G, denoted by rvc(G). is the smallest positive integer k so that G has a rainbow vertex k-coloring. The distance between two difference vertices u and v in V(G), denoted by d(u,v), is the length of a shortest u-v path in G. For i ∈ {1,2,…, k], let Ri be the set of vertices with color i and Π - {R 1,R 2,…,Rk } be fjn ordered partition of V(G). The rainbow code of a vertex v of V(G) with respect to Π is defined as the k-tuple , where d(v, Ri ) =min{d(v, y) | y ∈ Ri } for each i ∈ {1,2,…, k}. If every vertex of G has distinct rainbow codes, then c is called a locating rainbow k-coloring of G. The locating rainbow connection number of G. denoted by rvcl(G). is defined as the smallest positive integer k such that G has a locating rainbow k-coloring. In this paper, we provide the sharp upper and lower bounds for locating rainbow connection number of a graph. We also determine the locating rainbow connection number of some well-known classes of graphs.

A Study on Strong Rainbow Vertex-Connection in Some Classes of Generalized Petersen Graphs

Abstract In a vertex colored graph G, a rainbow path is defined as a path in which all the internal vertices get different colors. The graph G is called a strongly rainbow vertex-connected graph, if at least one shortest rainbow path exists between every pair of distinct vertices. The strong rainbow vertex-connection number, represented by srvc(G) is the fewest number of colors needed for strong rainbow vertex coloring of the graph G. This paper explores sharp upper bounds for the strong rainbow vertex-connection number of GP graphs P(n,k) for the cases when k|n and n = mk +1, m is a positive integer.

The rainbow vertex connection number of edge corona product graphs

Let G1, G2 be a special graphs with vertices of G1 1,2,…, n and edges of G1 1,2,… m. The generalized edge corona product of graphs G1 and G2, denoted by G1 ⋄ G1 is obtained by taking one copy of graph G1 and m copy of G2, thus for each edge ek = ij of G, joining edge between the two end-vertices i, j of ek and each vertex of the k-copy of G2. A rainbow vertex-coloring graph G where adjacent vertices u-v and its internal vertices have distinct colors. A path is called a rainbow path if no two verticess of the path have the same color. A rainbow vertex-connection number of graph G is minimun number of colors in graph G to connected every two distinct internal vertices u and v such that a graph G naturally rainbow vertex-connected, denoted by rvc (G). In this paper, we determine minimum integer for rainbow vertex coloring of edge corona product on cycle and path such as Pn ⋄ Pm, Pn ⋄ Cm, Cn ⋄ Pm, and Cn ⋄ Cm.

The Rainbow Vertex-Connection Number of Star Fan Graphs

A vertex-colored graph is said to be rainbow vertex-connected, if for every two vertices and in , there exists a path with all internal vertices have distinct colors. The rainbow vertex connection number of , denoted by is the smallest number of colors needed to make rainbow vertex connected. In this paper, we determine the rainbow vertex connection number of star fan graphs.

On the (Strong) Rainbow Vertex Connection of Graphs Resulting from Edge Comb Product

The vertex-colored graph G = (V, E) is said rainbow vertex-connected, if for every two vertices u and v in V, there is a u − v path with all internal vertices have distinct color. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors required in order to make graph G to be rainbow vertex-connected. If every two vertices u and v in V are connected by at least one shortest rainbow path, then G is strongly rainbow vertex-connected. The strong rainbow vertex- connection number, denoted by srvc(G), is the minimum number of colors required in order make graph G to be strongly rainbow vertex-connected. In this paper, we determine the rainbow vertex connection number and the strong rainbow vertex connection number of graphs which are resulted from edge comb product.

Rainbow connections in digraphs

An arc-coloured path in a digraph is rainbow if its arcs have distinct colours. A vertex-coloured path is vertex rainbow if its internal vertices have distinct colours. A totally-coloured path is total rainbow if its arcs and internal vertices have distinct colours. An arc-coloured (resp. vertex-coloured, totally-coloured) digraph D is rainbow connected (resp. rainbow vertex-connected, total rainbow connected) if any two vertices of D are connected by a rainbow (resp. vertex rainbow, total rainbow) path. The rainbow connection number (resp. rainbow vertex-connection number, total rainbow connection number) of a digraph D is the smallest number of colours needed to make D rainbow connected (resp. rainbow vertex-connected, total rainbow connected).In this paper, we study the rainbow connection, rainbow vertex-connection and total rainbow connection numbers of digraphs. We give some properties of these parameters and establish relations between them. The rainbow connection number and the rainbow vertex-connection number of a digraph D are both upper bounded by the order of D, while its total rainbow connection number is upper bounded by twice of its order. In particular, we prove that a digraph of order n has rainbow connection number n if and only if it is Hamiltonian and has three vertices with eccentricity n−1, that it has rainbow vertex-connection number n if and only if it has a Hamiltonian cycle C and three vertices with eccentricity n−1 such that no two of them are consecutive on C, and that it has total rainbow connection number 2n if and only if it has rainbow vertex-connection number n.

Rainbow vertex-connection and forbidden subgraphs

A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path connecting them. For a connected graph $G$, the \emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined as the minimum number of colors that are required to make $G$ rainbow vertex-connected. In this paper, we find all the families $\mathcal{F}$ of connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant $k_\mathcal{F}$ such that, for every connected $\mathcal{F}$-free graph $G$, $rvc(G)\leq diam(G)+k_\mathcal{F}$, where $diam(G)$ is the diameter of $G$.