Artikel HafidhyahDW M0114015

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  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 - n labelings. A generalized butterfly graph, BF , 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

  4n

n n

butterfly graph, BF , for n ≥ 2. The result is tes(BF ) = ⌈ ⌉.

  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. c

  Baˇ a 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 = u v and f = u v then their weight wt(e) ̸= wt(f ). The weight of an edge uv in G, i j k l 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 c a 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,

  Many researchers have investigated tes(G) to some graph classes. By then, there are some result related to the tes(G). Baˇ c a et al. [1] proved tes(G) of paths, cycles, stars, wheels, and n n

  • 2 +1 2n+2

  friendship graphs, that are, tes(P n ) = tes(C n ) = ⌈ ⌉, tes(S n ) = ⌈ ⌉, tes(W n ) = ⌈ ⌉ for

  3

  2

  3 3n+2 n ≥ 3, and tes(F ) = ⌈ ⌉. Then, tes(G) of tree G with edge set E and maximum degree n

  3 |E|+2

  △+1 △ had been found by Ivanˇ c o and Jendrol’ [6], that is tes(G) = max{⌈ ⌉, ⌈ ⌉}. In 2008,

  2

  3 Nurdin et al. [7] proved tes(G) of the corona product of paths with some graphs, namely paths P n , cycles C n , stars S n , gears G n , friendships F n , and wheels W n . Haque [3] in 2012 proved tes (G) of generalized Petersen graphs P (n, k) and Indriati et al. [4] found tes(G) of Helm, H n ,

  1 4n+2 and disjoint union of t isomorphic helms, tH n . In 2013 Indriati et al. [5] found tes(H ) = ⌈ ⌉, n

  3 (m+3)n+2 2 5n+2 m tes (H ) = ⌈ ⌉, and tes(H ) = ⌈ ⌉ for n ≥ 3 and m ≡ 0 (mod 3). n n

  3

  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, BF n , for n ≥ 2.

  2. Main Result A generalized butterfly graph, BF n , 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 BF n be V (BF n ) = {v i | i = 0, 1, 2, ..., 2n} and the edge set of BF n

  , v , v be E(BF n ) = {(v i i ) | i = 1, 2, ..., n − 1, n + 1, ..., 2n − 1} ∪ {(v i ) | i = 1, 2, ..., 2n}. Figure

  • 1

  1 illustrates the generalized butterfly graph BF n . Based on Figure 1, BF n has {v } as an apex, , v , v , ..., v , v , v , v , ..., v , v

  {v n } as vertices on right wing, and {v } as

  1

  2 3 (n−1) (n+1) (n+2) (n+3) (2n−1) 2n vertices on left wing.

  (n+1) v n v

  (n+2) v

  (n-1) v

v (n+3) (n-2)

v

v

  (2n-2) v

  3 v

  (2n-1) v

  2 v

  

2n v

  1 v Figure 1.

  The generalized butterfly graph BF n In the next theorem, we present the total edge irregularity strength of generalized butterfly graph, BF n , for n ≥ 2 as follows:

  4n Theorem 2.1 For n ≥ 2, tes(BF ) = ⌈ ⌉. n

  3 4n

  Proof . From the lower bound of total edge irregularity strength we have that tes(BF n ) ≥ ⌈ ⌉,

  3 n ≥ 2. To prove the equality, it is sufficient to show the existence of an edge irregular

  4n ⌉.

  1 (v i −1

  1 (v i v i

  3 − i(mod 2)), for n + 2 ≤ i ≤ 2n. f

  ( 2n+3

  −1 ) + (−1) i

  1 (v i

  ) + 1, for i = n + 1; f

  3 ≤ i ≤ n; f

  2n−3

  2n+6

  3 − i(mod 2)), for

  ( 2n−3

  −1 ) + (−1) i −1

  1 (v i

  3 ; f

  2n+3

  ) =                i, for 1 ≤ i ≤

  3 ;

  2n

  3 ;

  1 (v ) = ⌈

  Case 3: For n ≡ 1(mod 3), n ≥ 4. f

  3 ≤ i ≤ 2n.

  2n+6

  3 ⌉, for

  ⌈ 4n

  2n+3

  1, for i = 2n

  3

  2n−3

        

  1 (v v i ) =

  3 ≤ i ≤ n − 1; . f

  2, for 2n+3

  3 and n + 1 ≤ i ≤ 2n − 1;

  3 , for i =

  3 ;

  inequality, we define a function f

  3 ; f

  3 ≤ i ≤ 2n. f

  2n+8

  3 − i(mod 2)), for

  ( 2n−1

  −1 ) + (−1) i

  1 (v i

  1, for 1 ≤ i ≤ 2n+5

  ) =                i, for 1 ≤ i ≤

        

  1 (v i ) =

  3 ⌉. f

  4n

  1 (v ) = ⌈

  Case 1: For n ≡ 2(mod 3), n ≥ 2. f

  1 as follows.

  1 (v i v i +1

  2n+2

  1, for 1 ≤ i ≤ 2n

  3 ⌉, for

  1 (v i ) =                               

  3 ⌉. f

  4n

  1 (v ) = ⌈

  Case 2: For n ≡ 0(mod 3), n ≥ 3 and n is odd. f

  3 ≤ i ≤ 2n.

  2n+8

  4n

  3 ;

  3 ⌈

  3 2n+5

  2n−4

        

  1 (v v i ) =

  3 ≤ i ≤ n − 1; 1, for n + 1 ≤ i ≤ 2n − 1. f

  2, for 2n+5

  • 1
  • i, for 1 ≤ i ≤

   2n+7

  1, for 1 ≤ i ≤ ; 

  3      i 2n+1

    f (v ) + (−1) ( − i(mod 2)), for 6 ≤ i ≤ 8 and n = 4;

  1 i −1 

  3        f

  1 (v i −1 ) + 1, for i = 8 and n = 7;    f

  (v i ) = 1 i −1 2n−5 f (v ) + (−1) ( + i(mod 2)), for 9 ≤ i ≤ 14 and n = 7;

  3      

   1 i −1

  2n+10 2n+13  f

   (v i ) + 1, for ≤ i ≤ and n ≥ 10; 1 −1

  

  3

  3      i 2n−5 2n+16

    f 1 (v i −1 ) + (−1) ( − i(mod 2)), for ≤ i ≤ 2n and n ≥ 10. 

  3

  3  

  2n+4 i, for 1 ≤ i ≤ ;

  

  3       2n+4 2n+7 

  , for i = and n ≥ 10; 

  3

  3      f (v v ) − 1, for ≤ i ≤ n − 2 and n ≥ 13;

  1 i −1 i

  3 f v

  (v i i ) = 1 +1

       2, for i = n − 1 and n ≥ 16;         1, for n + 1 ≤ i ≤ 2n − 1.

    

  2n−5 2n+7

  • i, for 1 ≤ i ≤ ;

  

  3

  3       4n 2n+10

  ⌈ ⌉, for ≤ i ≤ 2n and n ≥ 7;

  3

  3 f v

  (v i ) =

  1   

  2n+10   5 + i(mod 2), for ≤ i ≤ 2n and n = 4; 

  3 

  4n It can be seen that the function f 1 is a map from V (BF n ) ∪ E(BF n ) into {1, 2, ..., ⌈ ⌉}.

  3 4n Therefore, f is a total k -labeling with k = ⌈ ⌉.

  1

  1

  1

3 We observe that the weights of the edges are:

   2 + i, for 1 ≤ i ≤ n − 1;

     wt v

  (v i i ) =

  • 1 1 + i, for n + 1 ≤ i ≤ 2n − 1.

     Case 1: For n ≡ 2(mod 3), n ≥ 2.

   2n+5

  2n + i, for 1 ≤ i ≤ ; 

  3   wt (v v ) = i i 2n−1 2n+8

   wt (v v i ) + (−1) ( − i(mod 2)), for ≤ i ≤ 2n.

  −1 

  3

  3 

  • i(mod 2)), for
  • i(mod 2)), for 9 ≤ i ≤ 14 and n = 7; wt
    • labeling, f

  1

  2

  2

  6

  5

  4

  3

  2

  6

  1

  3

  1

  1

  1

  1

  1

  1

  1

  1

  1

  7

  8

  1

  1

  11

  11

  11

  11

  11

  11

  11

  11

  1

  11

  41

  10

  9

  8

  7

  6

  5

  10

  5

  91

  1

  1

  Case 2: For n ≡ 0(mod 3), n ≥ 3 and n is odd. wt (v v i ) =

  ) + (−1) i

  ( 2n−2

  −1 ) + (−1) i

  (v v i

  3 ; wt

  2n + i, for 1 ≤ i ≤ 2n+7

  ) =                                       

  Case 3: For n ≡ 1(mod 3), n ≥ 4. wt (v v i

  3 − i(mod 2)), for n + 2 ≤ i ≤ 2n.

  ( 2n+3

  (v v i −1 ) + 1, for i = n + 1; wt (v v i −1

  ) + 1, for i = 8 and n = 7; wt (v v i

  3 ≤ i ≤ n; wt

  2n+3

  3

  2n−3

  −1 (

  ) + (−1) i

  3 ; wt (v v i −1

  2n + i, for 1 ≤ i ≤ 2n

                        

  3 − i(mod 2)), for 6 ≤ i ≤ 8 and n = 4; wt (v v i −1

  −1 ) + (−1) i −1

  1

  3 ≤ i ≤ 2n and n ≥ 10.

  8 .

  Figure 2 shows an edge irregular total labeling of BF

  3 ⌉, for n ≥ 2. ⊓ ⊔

  4n

  3 ⌉. Therefore, tes(BF n ) = ⌈

  1 -labeling with k 1 = ⌈ 4n

  1 , form consecutive integers from 3 to 4n. It means that the weights of all edges are distinct, then we have tes(BF n ) ≤ k 1 . This completes the proof. So, the labeling is an edge irregular total k

  1

  It can be seen that the weights of edges of BF n under the total k

  2n+16

  ( 2n−5

  3 − i(mod 2)), for

  ( 2n−5

  −1 ) + (−1) i

  3 and n ≥ 10; wt (v v i

  2n+13

  3 ≤ i ≤

  2n+10

  (v v i −1 ) + 1, for

  3

  

11

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 BF n 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

  c

[1] Baˇ a 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 c o 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 a

  [8] Wallis W D. Magic Graphs 2001 Boston: Birkh¨ user

[9] Weisstein, Eric W. Butterfly Graph MathWorld, From MathWorld-A Wolfram Web Resource

http://mathworld.wolfram.com/ButterflyGraph.html.

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