On the Total Edge Irregularity Strength of
Generalized Butterfly Graph
Hafidhyah Dwi Wahyuna and Diari Indriati
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret, Surakarta, Indonesia
E-mail: dwhafidhyah@gmail.com, diari indri@yahoo.co.id
Abstract. LetG(V, E) be a connected, simple, and undirected graph with vertex setV and edge setE. A totalk-labeling is a map that carries vertices and edges of a graphGinto a set of positive integer labels {1,2, ..., k}. An edge irregular total k-labeling λ: V(G)∪E(G) → {1,2, ..., k}of a graphGis a totalk-labeling such that the weights calculated for all edges are distinct. The weight of an edgeuvinG, denoted bywt(uv), is defined as the sum of the label of u, the label ofv, and the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label k over all such edge irregular total k -labelings. A generalized butterfly graph,BFn, obtained by inserting vertices to every wing with
assumption that sum of inserting vertices to every wing are same then it has 2n+ 1 vertices and 4n−2 edges. In this paper, we investigate the total edge irregularity strength of generalized butterfly graph,BFn, forn≥2. The result istes(BFn) =⌈
4n
3 ⌉.
1. Introduction
Let G(V, E) be a connected, simple, and undirected graph with vertex set V and edge set E. A labeling of a graph Gis a mapping that carries a set of graph elements into a set of positive integers, called labels (Wallis [8]). If the domain of mapping is a vertex set, or an edge set, or a union of vertex and edge sets, then the labeling is called vertex labeling, edge labeling, or total labeling, respectively. In Gallian’s survey [2], he showed that there were various kinds of labelings on graphs, and one of them was an irregular total labeling.
Baˇca et al. [1] introduced the notion of a total k-labeling, an edge irregular total k-labeling, and a vertex irregular total k-labeling. For a graph G(V, E), they defined a labeling f :V ∪E → {1,2, ..., k}to be a total k-labeling. An edge irregular total k-labeling with vertex set V and edge set E is a labeling λ: V(G)∪E(G) → {1,2, ..., k} such that for two different edges e=uivj and f =ukvl then their weight wt(e) ̸=wt(f). The weight of an edge uv inG, denoted by wt(uv), is defined as the sum of the label of u, the label of v, and the label of uv, that is wt(uv) = λ(u) +λ(uv) +λ(v).
Furthermore, Baˇca et al. [1] also defined the total edge irregularity strength of G, denoted by tes(G), as the minimum value of the largest labelkover all such edge irregular totalk-labelings. They also gave a lower bound and an upper bound ontes(G) with vertex setV and a non-empty edge set E,
⌈|E|+ 2
3
⌉
Many researchers have investigated tes(G) to some graph classes. By then, there are some result related to the tes(G). Baˇca et al. [1] proved tes(G) of paths, cycles, stars, wheels, and friendship graphs, that are, tes(Pn) =tes(Cn) =⌈n+23 ⌉,tes(Sn) =⌈n+12 ⌉,tes(Wn) =⌈2n3+2⌉for n ≥ 3, and tes(Fn) = ⌈3n3+2⌉. Then, tes(G) of tree G with edge set E and maximum degree
△ had been found by Ivanˇco and Jendrol’ [6], that is tes(G) =max{⌈△+12 ⌉,⌈|E|+23 ⌉}. In 2008, Nurdin et al. [7] proved tes(G) of the corona product of paths with some graphs, namely paths Pn, cycles Cn, stars Sn, gears Gn, friendships Fn, and wheels Wn. Haque [3] in 2012 proved tes(G) of generalized Petersen graphsP(n, k) and Indriatiet al. [4] foundtes(G) of Helm,Hn, and disjoint union oftisomorphic helms,tHn. In 2013 Indriatiet al. [5] foundtes(Hn1) =⌈4n3+2⌉, tes(Hn2) =⌈5n3+2⌉, andtes(Hm
n) =⌈
(m+3)n+2
3 ⌉ forn≥3 andm≡0 (mod 3).
Weisstein [9] defined the butterfly graph is a planar undirected graph with 5 vertices and 6 edges. In this paper, we investigate the total edge irregularity strength of generalized butterfly graph, BFn, for n≥2.
2. Main Result
A generalized butterfly graph,BFn, obtained by inserting vertices to every wing with assumption that sum of inserting vertices to every wing are same then it has 2n+ 1 vertices and 4n−2 edges. Let the vertex set of BFn beV(BFn) ={vi |i= 0,1,2, ...,2n} and the edge set ofBFn beE(BFn) ={(vi, vi+1)|i= 1,2, ..., n−1, n+ 1, ...,2n−1} ∪ {(v0, vi)|i= 1,2, ...,2n}. Figure 1 illustrates the generalized butterfly graphBFn. Based on Figure 1,BFn has{v0}as an apex,
{v1, v2, v3, ..., v(n−1), vn} as vertices on right wing, and {v(n+1), v(n+2), v(n+3), ..., v(2n−1), v2n} as vertices on left wing.
v2
v3
v(n-2)
v(n-1)
vn
v(n+1)
v(n+2)
v(n+3)
v(2n-2)
v(2n-1)
v2n
v0
v1
Figure 1. The generalized butterfly graphBFn
In the next theorem, we present the total edge irregularity strength of generalized butterfly graph, BFn, for n≥2 as follows:
Theorem 2.1 For n≥2, tes(BFn) =⌈43n⌉.
f1(vi) =
We observe that the weights of the edges are:
Case 2: For n≡0(mod 3), n≥3 and n is odd.
It can be seen that the weights of edges of BFn under the total k1-labeling, f1, form consecutive integers from 3 to 4n. It means that the weights of all edges are distinct, then we have tes(BFn) ≤ k1. This completes the proof. So, the labeling is an edge irregular total k1-labeling withk1 =⌈43n⌉. Therefore,tes(BFn) =⌈43n⌉, forn≥2. ⊓⊔
Figure 2 shows an edge irregular total labeling ofBF8.
1
3. Concluding Remark
Furthermore, we conclude this paper with the following open problem for the direction of further research which is still in progress.
Open Problem: Determine the total edge irregularity strength of generalized butterfly graph BFn for n ≥ 3 and n is odd. Then, generalized butterfly graph isomorphic with broken fan graph. Broken fan graph is fan graph which divide into n parts for n = 2. So, determine the total edge irregularity strength of broken fan graph which divide into nparts forn≥3.
Acknowledgments
We gratefully acknowledge the support from Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret Surakarta. Then, we wish to thank the referees for their valuable suggestions and references, which helped to improve the paper.
References
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#DS6
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