J. Comp. & Math. Sci. Vol.2 (3), 505-513 (2011)
Edge Bimagic Total Labelings of Certain Graphs M. A. BASKER Department of Mathematics, Loyola College, Chennai-600034, India. ABSTRACT A ( p , q ) -graph
G is said to have an edge bimagic total labeling
if there exists a bijection
f : V ∪ E → {1,2 ... p + q} such that
for each edge e = uv ∈ E (G ) , where
f (u) + f (e) + f (v) = k1 or k2 ,
k1 and k2 are two distinct constants called magic constants.
In this paper we identify classes of graphs which are edge bimagic total. Keywords: Labeling, Edge-magic total labeling.
1. INTRODUCTION A graph labeling is an assignment of integers to elements of a graph: the vertices or edges, or both, subject to certain conditions. Labeled graphs serve as useful models for a broad range of applications such as coding theory, x-ray, crystallography, radar, astronomy, circuit design, communication network addressing and data base management. In 1970, Kotzig and Rosa11 defined an edge-magic total labeling of a ( p , q ) graph G as a bijection f from V ∪ E to {1, 2, ... p + q} such that for all edges xy, the number f(x) + f(y) + f(xy) is constant. An edge-magic total labeling is called super edge-magic total if the set of vertex labels is {1, 2, ... p} . Enomoto et al.5 proved that Cn is super edge-magic total if and only if n is odd; all caterpillars are super edge-magic
total; K m, n is super edge-magic total if and only if m = 1 or n = 1 and Kn is super edgemagic total if and only if n = 1, 2 or 3. They also proved that if a (p, q)-graph G is super edge-magic total then q ≤ 2 p − 3 and they conjecture that every tree is super edgemagic total. Lee and Shan12 have verified this conjecture for trees with upto 17 vertices, using computer simulation. The concept of edge bimagic labelings introduced by Babujee3 is taken up for investigation in this paper. A ( p , q ) graph G is said to have an edge bimagic total labeling if there exists a bijection {1, 2, ... p + q} such that for each edge
e = uv ∈ E(G) , f (u) + f (e) + f (v) = k1 or k2 , where k1 and k2 are two distinct constants called magic constants. It is known that Wn has no edge magic total labeling when
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M. A. Basker, J. Comp. & Math. Sci. Vol.2 (3), 505-513 (2011)
n ≡ 3(mod 4) . Marr et al.[1] have exhibited edge bimagic total labeling for wheels Wn , when n ≡ 3(mod 4) . They have also
three vertices v11 , v12 , v13 of Pn × C3 . We denote this graph by PY (n) . This graph has
3n + 1 vertices and 6n edges. See Figure 2.
constructed edge bimagic total labeling for one-factors and complete graphs K 4 (= W3 ) ,
K7 and K8 . The graph in the Figure 1 is edge bimagic total with magic constants 27 and 28. 8
12 15 11
24
1
16
17 2
18
22
10
7
23 4
13
14 9
20
3 21
5
19 6
Figure 1: An example for edge bimagic total
In this paper we identify three classes of graphs which are edge bimagic total. 2. Pyramid Graphs Prisms are graphs of the form Pn × Cm . These can be viewed as grids on cylinders of height n − 1 . In this paper we construct a graph, called a pyramid, from the prism Pn × C3 . Let
V ( Pn × C3 ) = {vij :1 ≤ i ≤ n,1 ≤ j ≤ 3} .
A pyramid graph is obtained from Pn × C3 by adding a new vertex v00 adjacent to the
Figure 2: The pyramid graph PY (4)
Theorem 1 Let G be PY (n) . Then G is edge bimagic total for n ≥ 3 . Proof. Let f : V ∪ E → {1, 2 ... 9n + 1} be a bijection defined by f (v00 ) = 1 ,
f (v00 v11 ) = 9n + 1 , f (v00v12 ) = 9n , f (v00 v13 ) = 9n − 1 , f (v3i − 2,1 ) = 9i − 7 , f ( v3i − 2,2 ) = 9i − 6 , f (v3i − 2,3 ) = 9i − 5 , f (v3 j −1,1 ) = 9 j − 2 , f (v3 j −1,2 ) = 9 j − 4 , f (v3 j −1,3 ) = 9 j − 3 , f ( v3 k ,1 ) = 9 k , f ( v3 k ,2 ) = 9 k + 1 , f (v3 k ,3 ) = 9 k − 1 , f (v3i − 2,1v3i − 2,2 ) = 9 n − 18i + 16 , f ( v3i − 2,2 v3i − 2,3 ) = 9 n − 18i + 14 , f ( v3i − 2,3 v3i − 2,1 ) = 9 n − 18i + 15 , f (v3 j −1,1v3 j −1,2 ) = 9 n − 18 j + 9 ,
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
M. A. Basker, J. Comp. & Math. Sci. Vol.2 (3), 505-513 (2011)
f ( v3 j −1,2 v3 j −1,3 ) = 9 n − 18 j + 10 , f (v3 j −1,3v3 j −1,1 ) = 9 n − 18 j + 8 , f (v3 k ,1v3 k ,2 ) = 9 n − 18 j + 2 , f (v3 k ,2 v3 k ,3 ) = 9 n − 18 k + 3 , f ( v3 k ,3 v3 k ,1 ) = 9 n − 18k + 4 , where
507
n ≡ 0(mod 3) (b) i = 1, 2... n / 3 + 1; j , k = 1, 2... n / 3 , when n ≡ 1(mod 3) (c) i, j = 1, 2... n / 3 + 1; k = 1, 2... n / 3 , when n ≡ 2(mod 3) .
(a) i, j , k = 1, 2... n / 3 , when
Figure 3: Vertices of PY ( n ) at consecutive levels
f (v3i − 2,1v3i −1,1 ) = 9n − 18i + 12 , f (v3 j −1,1v3 j ,1 ) = 9n − 18 j + 5 , f (v3k ,1v3( k +1)−2,1 ) = 9n − 18k + 1 , f (v3i−2,2v3i −1,2 ) = 9n −18i + 13 , f (v3 j −1,2 v3 j ,2 ) = 9n − 18 j + 6 , f ( v3 k ,2 v3( k +1) − 2,2 ) = 9 n − 18k − 1 , f (v3i − 2,3v3i −1,3 ) = 9 n − 18i + 11 , f (v3 j −1,3v3 j ,3 ) = 9 n − 18 j + 7 , f (v3 k ,3 v3( k +1) − 2,3 ) = 9 n − 18k , where Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
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M. A. Basker, J. Comp. & Math. Sci. Vol.2 (3), 505-513 (2011)
(a)
i, j = 1, 2... n / 3 ; k = 1, 2... n / 3 − 1 , when n ≡ 0(mod 3)
(b)
i, j , k = 1, 2... n / 3 , when n ≡ 1(mod 3)
(c)
i = 1, 2... n / 3 + 1; j , k = 1, 2... n / 3 , when n ≡ 2(mod 3) .
Figure 4: Labeling of the graph PY ( n ) in the case i = j = k
We shall prove that this labeling is edge bimagic total. Now f (v00 ) + f (v11 ) + f (v00 v11 ) = 1 + 2 + 9n + 1 = 9n + 4 . Similarly f (v00 ) + f (v12 ) + f (v00 v12 ) = f (v00 ) + f (v13 ) + f (v00 v13 ) = 9n + 4 . Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
M. A. Basker, J. Comp. & Math. Sci. Vol.2 (3), 505-513 (2011)
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Figure 5: Edge bimagic total labeling of PY (4)
Given n , considering appropriate values for i , j and k , we have
f ( v3i − 2,1 ) + f (v3i − 2,2 ) + f (v3i − 2,1v3i − 2,2 ) = 9i − 7 + 9i − 6 + 9 n − 18i + 16 = 9 n + 3 , f ( v3 j −1,1 ) + f (v3 j −1,2 ) + f ( v3 j −1,1v3 j −1,2 ) = 9 j − 2 + 9 j − 4 + 9 n − 18 j + 9 = 9 n + 3 and f (v3 k ,1 ) + f ( v3 k ,2 ) + f (v3 k ,1v3 k ,2 ) = 9 k + 9 k + 1 + 9 n − 18k + 2 = 9 n + 3 . It is easy to check that for any edge
uv , the sums '' f (u ) + f (v ) + f (uv ) '' equal
9n + 3 . See Figures 3 and 4. 3. SNAKE GRAPHS The motivation for the classes of graphs considered in this section is the class of triangular ular snakes introduced by Rosa11. A triangular snake is a connecte connected graph all of whose blocks are triangles and whose block blockcutpoint graph is a path. We consider a class
of snakes all of whose blocks are K 4 , the complete graphs on 4 vertices. We denote this class by nK 4 and prove that nK 4 is edge bimagic total. The vertex set of nK 4 is
{xi :1 ≤ i ≤ n + 1} ∪ { yi , wi :1 ≤ i ≤ n} and the edge set is
{ xi xi +1 , xi yi , yi xi +1 , xi wi , wi xi +1 :1 ≤ i ≤ n} .
See Figure 6.
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M. A. Basker, J. Comp. & Math. Sci. Vol.2 (3), 505-513 (2011)
Figure 6: The graph 4k4
Figure 7 : Labeling of the k th block of nk4
Theorem 2 The graph nK 4 is edge bimagic total for n ≥ 1. Proof. Consider the k th block of nK 4 and label the vertices and edges as follows:
f ( xk ) = 3k − 2 , f ( xk +1 ) = 3k + 1 , f ( yk ) = 3k , f ( wk ) = 3k − 1 and f ( xk xk +1 ) = 9n + 5 − 6k , f ( xk yk ) = 9n + 6 − 6k , f ( yk xk +1 ) = 9n + 2 − 6k , f ( xk wk ) = 9n + 7 − 6k , f ( wk xk +1 ) = 9n + 3 − 6k . See Figure 7. It is sufficient to prove that the k th block of nK 4 is edge bimagic total,
1 ≤ k ≤ n . f ( xk ) + f ( xk +1 ) + f ( xk xk +1 ) = 3k − 2 + 3k + 1 + 9n + 5 − 6k = 9n + 4 , f ( yk ) + f ( xk +1 ) + f ( yk xk +1 ) = 3k + 3k + 1 + 9n + 2 − 6k = 9n + 3 . Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580))
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Figure 8 : Example to illustrate Theorem 2
It can be verified that for any edge uv , the expression '' f (u ) + f (v ) + f (uv ) '' yields either 9n + 4 or 9n + 3 . We consider another class of snakes denoted nW4 , all of whose blocks are isomorphic to the wheel W4 and whose block block-cutpoint graph is a path.
Figure 9 : The snake nW4
The vertex set of nW4 is { xi :1 ≤ i ≤ n + 1} ∪ { yi , zi , wi :1 ≤ i ≤ n} and edge set is
{ xi yi , xi zi , xi wi , zi xi +1 , yi xi +1 , wi xi +1 :1 ≤ i ≤ n} . See Figure 9. Theorem 3 The graph nW4 is edge bimagic total for n ≥ 1.
Proof. Consider the k th block of nW4 and label the vertices and edges as follows:
f ( xk ) = 4k − 3 , f ( yk ) = 4k − 1 , f ( xk +1 ) = 4k + 1 , f ( zk ) = 4k − 2 , f ( wk ) = 4k , Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580))
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f ( xk yk ) = 12n − 8k + 8 , f ( yk xk +1 ) = 12n − 8k + 3 , f ( xk zk ) = 12n − 8k + 9 , f ( xk wk ) = 12n − 8k + 6 , f ( zk xk +1 ) = 12n − 8k + 5 , f ( wk xk +1 ) = 12n − 8k + 2 , f ( zk yk ) = 12n − 8k + 7 , f ( yk wk ) = 12n − 8k + 4 . See Figure 10.
Figure 10 : The k th block of nW4 and its labeling
To settle the claim it is sufficient to prove that the k th block of nW4 is edge bimagic total,
1 ≤ k ≤ n . Consider f ( xk ) + f ( yk ) + f ( xk yk ) = 4k − 3 + 4k −1 + 12n − 8k + 8 = 12n + 4 and f ( yk ) + f ( xk +1 ) + f ( yk xk +1 ) = 4k − 1 + 4k + 1 + 12n − 8k + 3 = 12n + 3 .
Figure 11: Example to illustrate Theorem 3
Similarly it can be shown that for any edge uv , the expression '' f (u ) + f (v ) + f (uv ) '' equals either 12n + 3 or 12n + 4 . See Figure 11. Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580))
M. A. Basker, J. Comp. & Math. Sci. Vol.2 (3), 505-513 (2011)
4. CONCLUSION In this paper we have constructed graphs which are edge bimagic total. The classes are derived from the prisms and snake graphs. It is interesting to identify other classes of graphs which are edge bimagic total. Two magic constants k1 and
8.
9.
k2 are involved in edge bimagic total labeling of graphs. For the graphs considered in this paper k1 − k 2 = 1 . It would be an interesting problem to construct such graphs. REFERENCES 1. Alison Marr, N. C. K. Philips, W. D. Wallis, Bimagic Labelings, Preprint. 2. M. Baca, On Magic Labelings of mPrisms, Math. Slovaca, 40, pages 11-14, (1990). 3. J. Basker Babujee, On Edge Bimagic Labeling, Journal of Combinatorics, Information & System sciences, 28, Nos. 1-4, pages 239-244, (2004). 4. Z. Chen, On Super Edge-Magic Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 38, pages 53-64, (2001). 5. H. Enomoto, A.S.Llado, T.Nakamigawa, and G.Ringel, Super Edge-Magic Graphs, SUT J. Math., 34, pages 105109, (1998). 6. R. M. Figueroa-Centeno, R. Ichishima, and F. Muntaner-Batle, On Edge-Magic Labelings of Certain Disjoint Unions of Graphs, preprint. 7. R. Figueroa-Centeno, R. Ichishima, and
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