Artikel HafidhyahDW M0114015
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. Let G(V, E) be a connected, simple, and undirected graph with vertex set V and
edge set E. A total k -labeling is a map that carries vertices and edges of a graph G into 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 graph G is a total k -labeling such that the weights calculated for all edges are
distinct. The weight of an edge uv in G, denoted by wt(uv), is defined as the sum of the label
of u, the label of v, 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 , for n ≥ 2. The result is tes(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 G is 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.
Baˇ
ca 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 = ui vj and f = uk vl then their weight wt(e) ̸= wt(f ). The weight of an edge uv in G,
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, Baˇ
ca et al. [1] also defined the total edge irregularity strength of G, denoted by
tes(G), as the minimum value of the largest label k over all such edge irregular total k -labelings.
They also gave a lower bound and an upper bound on tes(G) with vertex set V and a non-empty
edge set E,
⌉
⌈
|E| + 2
≤ tes(G) ≤ |E|.
(1)
3
Many researchers have investigated tes(G) to some graph classes. By then, there are some
result related to the tes(G). Baˇ
ca et al. [1] proved tes(G) of paths, cycles, stars, wheels, and
n+1
2n+2
friendship graphs, that are, tes(Pn ) = tes(Cn ) = ⌈ n+2
3 ⌉, tes(Sn ) = ⌈ 2 ⌉, tes(Wn ) = ⌈ 3 ⌉ for
3n+2
n ≥ 3, and tes(Fn ) = ⌈ 3 ⌉. Then, tes(G) of tree G with edge set E and maximum degree
|E|+2
△ had been found by Ivanˇ
co and Jendrol’ [6], that is tes(G) = max{⌈ △+1
2 ⌉, ⌈ 3 ⌉}. 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 graphs P (n, k) and Indriati et al. [4] found tes(G) of Helm, Hn ,
and disjoint union of t isomorphic helms, tHn . In 2013 Indriati et al. [5] found tes(Hn1 ) = ⌈ 4n+2
3 ⌉,
(m+3)n+2
5n+2
m
2
⌉ for n ≥ 3 and m ≡ 0 (mod 3).
tes(Hn ) = ⌈ 3 ⌉, and tes(Hn ) = ⌈
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 be V (BFn ) = {vi | i = 0, 1, 2, ..., 2n} and the edge set of BFn
be E(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 graph BFn . 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.
v(n+1)
vn
v(n+2)
v(n-1)
v(n+3)
v(n-2)
v0
v(2n-2)
v3
v(2n-1)
v2
v2n
v1
Figure 1. The generalized butterfly graph BFn
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 ) = ⌈ 4n
3 ⌉.
Proof. From the lower bound of total edge irregularity strength we have that tes(BFn ) ≥ ⌈ 4n
3 ⌉,
n ≥ 2. To prove the equality, it is sufficient to show the existence of an edge irregular
4n
total k1 -labeling with k1 = ⌈ 4n
3 ⌉. Let k1 = ⌈ 3 ⌉. Then from inequality (1) it follows that,
tes(BFn ) ≥ ⌈ |E(BF3n )|+2 ⌉ = ⌈ (4n−2)+2
⌉ = ⌈ 4n
3
3 ⌉ = k1 , that is tes(BFn ) ≥ k1 . To prove the reverse
inequality, we define a function f1 as follows.
Case 1: For n ≡ 2(mod 3), n ≥ 2.
4n
⌉.
3
1,
f1 (v0 ) = ⌈
for 1 ≤ i ≤
f1 (vi ) =
f (v ) + (−1)i ( 2n−1
3 − i(mod 2)),
1 i−1
i, for 1 ≤ i ≤ 2n+2
3 ;
2n+5
2, for
f1 (vi vi+1 ) =
1,
f1 (v0 vi ) =
for
2n+8
3
2n+5
3 ;
≤ i ≤ 2n.
≤ i ≤ n − 1;
3
for n + 1 ≤ i ≤ 2n − 1.
2n−4
3
+ i, for 1 ≤ i ≤
⌈ 4n ⌉,
3
for
2n+8
3
2n+5
3 ;
≤ i ≤ 2n.
Case 2: For n ≡ 0(mod 3), n ≥ 3 and n is odd.
4n
⌉.
3
1,
f1 (v0 ) = ⌈
f1 (vi ) =
for 1 ≤ i ≤
2n
3 ,
for i =
f1 (vi−1 ) + (−1)i−1 ( 2n−3
3 − i(mod 2)), for
f1 (vi−1 ) + 1,
i 2n+3
f1 (vi−1 ) + (−1) ( 3 − i(mod 2)),
f1 (vi vi+1 ) =
i,
1, for i =
2,
f1 (v0 vi ) =
for 1 ≤ i ≤
for
2n−3
3
2n+3
3
⌈ 4n ⌉,
3
4n
⌉.
3
for
2n+6
3
2n+3
3 ;
≤ i ≤ 2n.
2n+6
3
≤ i ≤ n;
for n + 2 ≤ i ≤ 2n.
and n + 1 ≤ i ≤ 2n − 1;
≤ i ≤ n − 1; .
2n+3
3 ;
for i = n + 1;
2n−3
3 ;
+ i, for 1 ≤ i ≤
Case 3: For n ≡ 1(mod 3), n ≥ 4.
f1 (v0 ) = ⌈
2n
3
2n
3 ;
f1 (vi ) =
f1 (vi vi+1 ) =
1,
f1 (vi−1 ) + (−1)i ( 2n+1
3 − i(mod 2)),
f (v ) + 1,
1 i−1
f1 (vi−1 ) + (−1)i−1 ( 2n−5
3 + i(mod 2)),
f1 (vi−1 ) + 1,
i 2n−5
f1 (vi−1 ) + (−1) ( 3 − i(mod 2)),
i,
2n+4
3 ,
f1 (vi−1 vi ) − 1, for
2,
1,
f1 (v0 vi ) =
2n+7
3
2n+10
3
2n+7
3 ;
for 6 ≤ i ≤ 8 and n = 4;
for i = 8 and n = 7;
for 9 ≤ i ≤ 14 and n = 7;
2n+13
3
for
2n+10
3
≤i≤
for
2n+16
3
≤ i ≤ 2n and n ≥ 10.
and n ≥ 10;
2n+4
3 ;
for 1 ≤ i ≤
for i =
for 1 ≤ i ≤
and n ≥ 10;
≤ i ≤ n − 2 and n ≥ 13;
for i = n − 1 and n ≥ 16;
for n + 1 ≤ i ≤ 2n − 1.
2n−5
3
+ i,
⌈ 4n
3 ⌉,
5 + i(mod 2),
for 1 ≤ i ≤
2n+7
3 ;
for
2n+10
3
≤ i ≤ 2n and n ≥ 7;
for
2n+10
3
≤ i ≤ 2n and n = 4;
It can be seen that the function f1 is a map from V (BFn ) ∪ E(BFn ) into {1, 2, ..., ⌈ 4n
3 ⌉}.
4n
Therefore, f1 is a total k1 -labeling with k1 = ⌈ 3 ⌉.
We observe that the weights of the edges are:
wt(vi vi+1 ) =
2 + i,
1 + i,
for 1 ≤ i ≤ n − 1;
for n + 1 ≤ i ≤ 2n − 1.
Case 1: For n ≡ 2(mod 3), n ≥ 2.
wt(v0 vi ) =
2n + i,
wt(v0 vi−1 ) + (−1)i ( 2n−1
3 − i(mod 2)),
for 1 ≤ i ≤
for
2n+8
3
2n+5
3 ;
≤ i ≤ 2n.
Case 2: For n ≡ 0(mod 3), n ≥ 3 and n is odd.
wt(v0 vi ) =
2n + i,
wt(v0 vi−1 ) + (−1)i−1 ( 2n−3
3 + i(mod 2)),
for 1 ≤ i ≤
for
wt(v0 vi−1 ) + 1,
i 2n+3
wt(v0 vi−1 ) + (−1) ( 3 − i(mod 2)),
2n+3
3
2n
3 ;
≤ i ≤ n;
for i = n + 1;
for n + 2 ≤ i ≤ 2n.
Case 3: For n ≡ 1(mod 3), n ≥ 4.
wt(v0 vi ) =
2n + i,
wt(v0 vi−1 ) + (−1)i ( 2n−2
3 − i(mod 2)),
wt(v0 vi−1 ) + 1,
for 1 ≤ i ≤
2n+7
3 ;
for 6 ≤ i ≤ 8 and n = 4;
for i = 8 and n = 7;
wt(v0 vi−1 ) + (−1)i−1 ( 2n−5
3 + i(mod 2)),
wt(v0 vi−1 ) + 1,
i 2n−5
wt(v0 vi−1 ) + (−1) ( 3 − i(mod 2)),
for 9 ≤ i ≤ 14 and n = 7;
2n+13
3
for
2n+10
3
≤i≤
for
2n+16
3
≤ i ≤ 2n and n ≥ 10.
and n ≥ 10;
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
4n
⊓
⊔
k1 -labeling with k1 = ⌈ 4n
3 ⌉. Therefore, tes(BFn ) = ⌈ 3 ⌉, for n ≥ 2.
Figure 2 shows an edge irregular total labeling of BF8 .
6
2
1
7
1
2
11
1
11
11
3
11
11
1
8
1
41
5
10
11
11
6
1
1
11
9
4
11
8
1
11
7
1
11
6
3
1
91
1
5
5
2
1
1
1
10
1
Figure 2. An edge irregular total 11-labeling of BF8
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 n parts for n ≥ 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
[1] Baˇ
ca M, Jendrol’ S, Miller M and Ryan J. On Irregular Total Labelings 2007 Discrete Math. 307 1378-1388
[2] Gallian J A. A Dynamic Survey of Graph Labeling 2016 The Electronic Journal of Combinatorics. 19 1-284
#DS6
[3] Haque K M M. Irregular Total Labellings of Generalized Petersen Graphs 2012 Theory Comput Syst. 50
537-544
[4] Indriati D, Widodo, Wijayanti I E and Sugeng K A. Kekuatan Tak Reguler Sisi Total pada Graf Helm
dan Union Disjoinnya 2012 Presented in Konferensi Nasional Matematika XVI, Unpad, Jatinangor, Jawa
Barat
[5] Indriati D, Widodo, Wijayanti I E and Sugeng K A. On the Total Edge Irregularity Strength of Generalized
Helm 2012 AKCE Int. J. Graphs Comb. 10 (2) 147-155
[6] Ivanˇ
co J and Jendrol’ S. Total Edge Irregularity Strength of Trees 2006 Discuss. Math. Graph Theory. 26
449-456
[7] Nurdin, Salman A N M and Baskoro E T. The Total Edge Irregularity Strength of the Corona Product of
Path with some Graphs 2013 J. Combin. Math. Combin. Comput. 65 163-175
[8] Wallis W D. Magic Graphs 2001 Boston: Birkh¨
auser
[9] Weisstein, Eric W. Butterfly Graph MathWorld, From MathWorld-A Wolfram Web Resource
http://mathworld.wolfram.com/ButterflyGraph.html.
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. Let G(V, E) be a connected, simple, and undirected graph with vertex set V and
edge set E. A total k -labeling is a map that carries vertices and edges of a graph G into 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 graph G is a total k -labeling such that the weights calculated for all edges are
distinct. The weight of an edge uv in G, denoted by wt(uv), is defined as the sum of the label
of u, the label of v, 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 , for n ≥ 2. The result is tes(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 G is 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.
Baˇ
ca 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 = ui vj and f = uk vl then their weight wt(e) ̸= wt(f ). The weight of an edge uv in G,
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, Baˇ
ca et al. [1] also defined the total edge irregularity strength of G, denoted by
tes(G), as the minimum value of the largest label k over all such edge irregular total k -labelings.
They also gave a lower bound and an upper bound on tes(G) with vertex set V and a non-empty
edge set E,
⌉
⌈
|E| + 2
≤ tes(G) ≤ |E|.
(1)
3
Many researchers have investigated tes(G) to some graph classes. By then, there are some
result related to the tes(G). Baˇ
ca et al. [1] proved tes(G) of paths, cycles, stars, wheels, and
n+1
2n+2
friendship graphs, that are, tes(Pn ) = tes(Cn ) = ⌈ n+2
3 ⌉, tes(Sn ) = ⌈ 2 ⌉, tes(Wn ) = ⌈ 3 ⌉ for
3n+2
n ≥ 3, and tes(Fn ) = ⌈ 3 ⌉. Then, tes(G) of tree G with edge set E and maximum degree
|E|+2
△ had been found by Ivanˇ
co and Jendrol’ [6], that is tes(G) = max{⌈ △+1
2 ⌉, ⌈ 3 ⌉}. 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 graphs P (n, k) and Indriati et al. [4] found tes(G) of Helm, Hn ,
and disjoint union of t isomorphic helms, tHn . In 2013 Indriati et al. [5] found tes(Hn1 ) = ⌈ 4n+2
3 ⌉,
(m+3)n+2
5n+2
m
2
⌉ for n ≥ 3 and m ≡ 0 (mod 3).
tes(Hn ) = ⌈ 3 ⌉, and tes(Hn ) = ⌈
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 be V (BFn ) = {vi | i = 0, 1, 2, ..., 2n} and the edge set of BFn
be E(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 graph BFn . 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.
v(n+1)
vn
v(n+2)
v(n-1)
v(n+3)
v(n-2)
v0
v(2n-2)
v3
v(2n-1)
v2
v2n
v1
Figure 1. The generalized butterfly graph BFn
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 ) = ⌈ 4n
3 ⌉.
Proof. From the lower bound of total edge irregularity strength we have that tes(BFn ) ≥ ⌈ 4n
3 ⌉,
n ≥ 2. To prove the equality, it is sufficient to show the existence of an edge irregular
4n
total k1 -labeling with k1 = ⌈ 4n
3 ⌉. Let k1 = ⌈ 3 ⌉. Then from inequality (1) it follows that,
tes(BFn ) ≥ ⌈ |E(BF3n )|+2 ⌉ = ⌈ (4n−2)+2
⌉ = ⌈ 4n
3
3 ⌉ = k1 , that is tes(BFn ) ≥ k1 . To prove the reverse
inequality, we define a function f1 as follows.
Case 1: For n ≡ 2(mod 3), n ≥ 2.
4n
⌉.
3
1,
f1 (v0 ) = ⌈
for 1 ≤ i ≤
f1 (vi ) =
f (v ) + (−1)i ( 2n−1
3 − i(mod 2)),
1 i−1
i, for 1 ≤ i ≤ 2n+2
3 ;
2n+5
2, for
f1 (vi vi+1 ) =
1,
f1 (v0 vi ) =
for
2n+8
3
2n+5
3 ;
≤ i ≤ 2n.
≤ i ≤ n − 1;
3
for n + 1 ≤ i ≤ 2n − 1.
2n−4
3
+ i, for 1 ≤ i ≤
⌈ 4n ⌉,
3
for
2n+8
3
2n+5
3 ;
≤ i ≤ 2n.
Case 2: For n ≡ 0(mod 3), n ≥ 3 and n is odd.
4n
⌉.
3
1,
f1 (v0 ) = ⌈
f1 (vi ) =
for 1 ≤ i ≤
2n
3 ,
for i =
f1 (vi−1 ) + (−1)i−1 ( 2n−3
3 − i(mod 2)), for
f1 (vi−1 ) + 1,
i 2n+3
f1 (vi−1 ) + (−1) ( 3 − i(mod 2)),
f1 (vi vi+1 ) =
i,
1, for i =
2,
f1 (v0 vi ) =
for 1 ≤ i ≤
for
2n−3
3
2n+3
3
⌈ 4n ⌉,
3
4n
⌉.
3
for
2n+6
3
2n+3
3 ;
≤ i ≤ 2n.
2n+6
3
≤ i ≤ n;
for n + 2 ≤ i ≤ 2n.
and n + 1 ≤ i ≤ 2n − 1;
≤ i ≤ n − 1; .
2n+3
3 ;
for i = n + 1;
2n−3
3 ;
+ i, for 1 ≤ i ≤
Case 3: For n ≡ 1(mod 3), n ≥ 4.
f1 (v0 ) = ⌈
2n
3
2n
3 ;
f1 (vi ) =
f1 (vi vi+1 ) =
1,
f1 (vi−1 ) + (−1)i ( 2n+1
3 − i(mod 2)),
f (v ) + 1,
1 i−1
f1 (vi−1 ) + (−1)i−1 ( 2n−5
3 + i(mod 2)),
f1 (vi−1 ) + 1,
i 2n−5
f1 (vi−1 ) + (−1) ( 3 − i(mod 2)),
i,
2n+4
3 ,
f1 (vi−1 vi ) − 1, for
2,
1,
f1 (v0 vi ) =
2n+7
3
2n+10
3
2n+7
3 ;
for 6 ≤ i ≤ 8 and n = 4;
for i = 8 and n = 7;
for 9 ≤ i ≤ 14 and n = 7;
2n+13
3
for
2n+10
3
≤i≤
for
2n+16
3
≤ i ≤ 2n and n ≥ 10.
and n ≥ 10;
2n+4
3 ;
for 1 ≤ i ≤
for i =
for 1 ≤ i ≤
and n ≥ 10;
≤ i ≤ n − 2 and n ≥ 13;
for i = n − 1 and n ≥ 16;
for n + 1 ≤ i ≤ 2n − 1.
2n−5
3
+ i,
⌈ 4n
3 ⌉,
5 + i(mod 2),
for 1 ≤ i ≤
2n+7
3 ;
for
2n+10
3
≤ i ≤ 2n and n ≥ 7;
for
2n+10
3
≤ i ≤ 2n and n = 4;
It can be seen that the function f1 is a map from V (BFn ) ∪ E(BFn ) into {1, 2, ..., ⌈ 4n
3 ⌉}.
4n
Therefore, f1 is a total k1 -labeling with k1 = ⌈ 3 ⌉.
We observe that the weights of the edges are:
wt(vi vi+1 ) =
2 + i,
1 + i,
for 1 ≤ i ≤ n − 1;
for n + 1 ≤ i ≤ 2n − 1.
Case 1: For n ≡ 2(mod 3), n ≥ 2.
wt(v0 vi ) =
2n + i,
wt(v0 vi−1 ) + (−1)i ( 2n−1
3 − i(mod 2)),
for 1 ≤ i ≤
for
2n+8
3
2n+5
3 ;
≤ i ≤ 2n.
Case 2: For n ≡ 0(mod 3), n ≥ 3 and n is odd.
wt(v0 vi ) =
2n + i,
wt(v0 vi−1 ) + (−1)i−1 ( 2n−3
3 + i(mod 2)),
for 1 ≤ i ≤
for
wt(v0 vi−1 ) + 1,
i 2n+3
wt(v0 vi−1 ) + (−1) ( 3 − i(mod 2)),
2n+3
3
2n
3 ;
≤ i ≤ n;
for i = n + 1;
for n + 2 ≤ i ≤ 2n.
Case 3: For n ≡ 1(mod 3), n ≥ 4.
wt(v0 vi ) =
2n + i,
wt(v0 vi−1 ) + (−1)i ( 2n−2
3 − i(mod 2)),
wt(v0 vi−1 ) + 1,
for 1 ≤ i ≤
2n+7
3 ;
for 6 ≤ i ≤ 8 and n = 4;
for i = 8 and n = 7;
wt(v0 vi−1 ) + (−1)i−1 ( 2n−5
3 + i(mod 2)),
wt(v0 vi−1 ) + 1,
i 2n−5
wt(v0 vi−1 ) + (−1) ( 3 − i(mod 2)),
for 9 ≤ i ≤ 14 and n = 7;
2n+13
3
for
2n+10
3
≤i≤
for
2n+16
3
≤ i ≤ 2n and n ≥ 10.
and n ≥ 10;
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
4n
⊓
⊔
k1 -labeling with k1 = ⌈ 4n
3 ⌉. Therefore, tes(BFn ) = ⌈ 3 ⌉, for n ≥ 2.
Figure 2 shows an edge irregular total labeling of BF8 .
6
2
1
7
1
2
11
1
11
11
3
11
11
1
8
1
41
5
10
11
11
6
1
1
11
9
4
11
8
1
11
7
1
11
6
3
1
91
1
5
5
2
1
1
1
10
1
Figure 2. An edge irregular total 11-labeling of BF8
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 n parts for n ≥ 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.
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