ON k-GRACEFUL LABELING OF SOME GRAPHS

Abstract

In this paper, it has been shown that the hairy cycle Cn ⊙ rK 1 , n ≡ 3(mod4), the graph obtained by adding pendant edge to each pendant vertex of hairy cycle Cn ⊙ 1K 1 , n ≡ 0(mod4), double graph of path Pn and double graph of comb Pn ⊙ 1K 1 are k-graceful.

An Analysis of the Factors Influencing the Strong Chromatic Index of Graphs Derived by Inflating a Few Common Classes of Graphs

“…(Figure 1) shows a comb graph (Figure 1a) and its inflated graph (Figure 1b). Recently, the labeling of the square graph of comb graphs [21] and k-graceful labeling of comb graphs [22] were discussed by the research community working on graph theory. We claim that the coloring scheme given in Table 1 colors all edges of the inflated comb graph G using only 6 colors.…”

LIFTS ON THE SUPERSTRUCTURE F(±a^2,±b^2) OBEYING (F^2+a^2)(F^2-a^2)(F^2+b^2)(F^2-b^2) = 0

“…The work of several scholars on various geometric structures and connections has been extremely beneficial, for instance, Saxena [6,7], Nivas et. Al [8,9], Mishra [10][11] Dida et.…”

Dividing Graceful Labeling of Certain Tree Graphs

“…A graph then can be thought of as drawing or diagram consisting of collection of vertices together with edges joining certain pairs of these vertices [1]. Let 𝐺(𝑉, 𝐸) be a simple undirected graph with order 𝑛 and size 𝑚 , if there exist an injective mapping 𝑓: 𝑉(𝐺) → {0,1,2,3, … , 𝑚} that induces a bijective mapping 𝑓 * : 𝐸(𝐺) → {1,2,3, … , 𝑚} defined by 𝑓 * (𝑢, 𝑣) = |𝑓(𝑢) − 𝑓(𝑣)| for each [𝑢, 𝑣] ∈ 𝐸(𝐺) and 𝑢 , 𝑣 ∈ 𝑉(𝐺), then 𝑓 is called graceful labeling of graph 𝐺 [2]. In 2002, Kourosh, E. Introduced a graceful with some operation graphs [1].…”

“…In 2012 Uma, R. and Murugesan, N. discussed graceful labeling of some simple graphs and their properties [3]. In 2014 Pradham, P. and Kumar, K. discussed the graphs which are obtained by adding the pendant edge to the vertices of 𝐾 𝑛 or 𝑃 2 or both in 𝑃 2 + 𝐾 𝑛 are graceful [2]. In 2014 , Munia, A. and et.…”

“…[2]: Let 𝐺 be a simple undirected graph with order 𝑛 and size 𝑚 , if there exist an injective mapping 𝑓: 𝑉(𝐺) → {0,1,2,3, … , 𝑚} that induces a bijective mapping 𝑓 * : 𝐸(𝐺) → {1,2,3, … , 𝑚} defined by 𝑓 * (𝑢, 𝑣) = |𝑓(𝑢) − 𝑓(𝑣)| for each (𝑢, 𝑣) ∈ 𝐸(𝐺) and 𝑢 , 𝑣 ∈ 𝑉(𝐺) , then 𝑓 is called graceful labeling of graph 𝐺 . Definition 2.2 [10]: A closed path is called a cycle .…”