On the total rainbow connection of the wheel related graphs

Eur. J. Pure Appl. Math.

et al. 2023

Given that a graph G = (V, E). By an edge-antimagic vertex labeling of graph, we mean assigning labels on each vertex under the label function f : V → {1, 2, . . . , |V (G)|} such that the associated weight of an edge uv ∈ E(G), namely w(xy) = f(x) + f(y), has distinct weight. A path P in the vertex-labeled graph G is said to be a rainbow path if for every two edges xy, x′y ′ ∈ E(P) satisfies w(xy) ̸= w(x ′y ′ ). The function f is called a rainbow antimagic labeling of G if for every two vertices x and y of G, there exists a rainbow x − y path. When we assign each edge xy with the color of the edge weight w(xy), thus we say the graph G admits a rainbow antimagic coloring. The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors induced from all edge weight of antimagic labeling. In this paper, we will study the rac(G) of the corona product of graphs. By the corona product of graphs G and H, denoted by G ⊙ H, we mean a graph obtained by taking a copy of graph G and n copies of graph H, namely H1, H2, ..., Hn, then connecting vertex vi from the copy of graph G to every vertex on graph Hi , i = 1, 2, 3, . . . , n. In this paper, we show the exact value of the rainbow antimagic connection number of Tn ⊙ Sm where Tn ∈ {Pn, Sn, Sn,p, Fn,3}.

Section: Introductionmentioning

confidence: 97%

On the study of Rainbow Antimagic Connection Number of Corona Product of Graphs

Septory

Susilowaty

Eur. J. Pure Appl. Math.

et al. 2023

“…The results of rainbow vertex coloring are found in [20], [22]. Another variant of rainbow coloring is total rainbow coloring and the results can be seen in [15] and [23]. Graph labeling was introduced in [24] by Wallis et al (2001).…”

Section: Introductionmentioning

confidence: 99%

On Rainbow Antimagic Coloring of Joint Product of Graphs

Septory

Susilowati

et al. 2023

CAUCHY

Let be a connected graph with vertex set and edge set . A bijection from to the set is a labeling of graph . The bijection is called rainbow antimagic vertex labeling if for any two edge and in path , where and . Rainbow antimagic coloring is a graph which has a rainbow antimagic labeling. Thus, every rainbow antimagic labeling induces a rainbow coloring G where the edge weight is the color of the edge . The rainbow antimagic connection number of graph is the smallest number of colors of all rainbow antimagic colorings of graph , denoted by . In this study, we studied rainbow antimagic coloring and have an exact value of rainbow antimagic connection number of joint product of graph where is graph , graph , graph , graph and graph .

“…A rainbow connection coloring is an edge-coloring in which G is rainbow connected. Rainbow vertex-connection is one of several interesting variants of the rainbow connection concept [5,6,9].…”

Section: Introductionmentioning

confidence: 99%

The rainbow vertex antimagic coloring of tree graphs

Marsidi¹,

Agustin

et al. 2022

J. Phys.: Conf. Ser.

Let G(V (G),E(G)) be a connected, simple, and finite graph. Let f be a bijective function of labeling on graph G from the edge set E(G) to natural number up to the number of edges of G. A rainbow vertex antimagic labeling of graph G is a function f under the condition all internal vertices of a path u – υ, Ɐu, υ ∈ V (G) have different weight (denoted by w(u)), where w(u) = ∑ uu′∈E(G)f (uu′). If G has a rainbow vertex antimagic labeling, then G is a rainbow vertex antimagic coloring, where the every vertex is assigned with the color w(u). The rvac(G) is a notation of rainbow vertex antimagic connection number of graph G which means the minimum colors taken over all rainbow vertex antimagic coloring induced by rainbow vertex antimagic labeling of graph G. The results of this research are the exact value of the rainbow vertex antimagic connection number of star (Sn ), double star (DSn ), and broom graph (Brn, m ).