Annexure
145 www.ccsenet.org/jmr Journal of Mathematics Research Vol. 3, No. 3; August 2011 Product Cordial Labeling in the Context of Tensor Product of Graphs
S K Vaidya (Corresponding author) Department of Mathematics, Saurashtra University Rajkot-360 005. GUJARAT, India E-mail: [email protected]
N B Vyas Atmiya Institute of Technology and Science Rajkot-360 005. GUJARAT, India E-mail: [email protected]
Received: March 10, 2011 Accepted: March 28, 2011 doi:10.5539/jmr.v3n3p83 Abstract
For the graph G1 and G2 the tensor product is denoted by G1(T p)G2 which is the graph with vertex set V(G1(T p)G2) = V(G1) × V(G2) and edge set E(G1(T p)G2) = {(u1, v1), (u2, v2)/u1u2E(G1) and v1v2E(G2)}. The graph Pm(T p)Pn is disconnected for ∀m, n while the graphs Cm(T p)Cn and Cm(T p)Pn are disconnected for both m and n even. We prove that these graphs are product cordial graphs. In addition to this we show that the graphs obtained by joining the connected components of respective graphs by a path of arbitrary length also admit product cordial labeling. Keywords: Cordial labeling, Porduct cordial labeling, Tensor product AMS Subject classification (2010): 05C78. 1. Introduction We begin with simple, finite and undirected graph G = (V(G), E(G)). For standard terminology and notations we follow (West, D. B, 2001). The brief summary of definitions and relevant results are given below. 1.1 Definition: If the vertices of the graph are assigned values subject to certain condition(s) then it is known as graph labeling. 1.2 Definition: A mapping f : V(G) →{0, 1} is called binary vertex labeling of G and f (v) is called the label of vertex v of G under f . ∗ ∗ For an edge e = uv, the induced edge labeling f : E(G) →{0, 1} is given by f (e = uv) = | f (u) − f (v)|. Let v f (0), v f (1) be the number of vertices of G having labels 0 and 1 respectively under f and let e f (0), e f (1) be the number of edges of G having labels 0 and 1 respectively under f ∗.
1.3 Definition: A binary vertex labeling of graph G is called a cordial labeling if |v f (0)−v f (1)|≤1 and |e f (0)−e f (1)|≤1. A graph G is called cordial if admits cordial labeling. The concept of cordial labeling was introduced by (Cahit,1987, p.201-207) and in the same paper he investigated several results on this newly introduced concept. Motivated through cordial labeling the concept of product cordial labeling was introduced in (Sundaram, M., Ponraj, R. and Somsundaram, S., 2004, p.155-163 ) which has the flavour of cordial lableing but absolute difference of vertex labels is replaced by product of vertex labels. 1.4 Definition: A binary vertex labeling of graph G with induced edge labeling f ∗ : E(G) →{0, 1} defined by ∗ f (e = uv) = f (u) f (v) is called a product cordial labeling if |v f (0) − v f (1)|≤1 and |e f (0) − e f (1)|≤1. A graph G is product cordial if it admits product cordial labeling. In (Sundaram, M., Ponraj, R. and Somsundaram, S., 2004, p.155-163) it has been proved that trees, unicyclic graphs of odd order, triangular snakes, dragons, helms and union of two path graphs are product cordial. They also proved that a ≥ < p2−1 graph with p vertices and q edges with p 4 is product cordial then q 4 . The graphs obtained by joining apex vertices of k copies of stars, shells and wheels to a new vertex are proved to be product cordial in (Vaidya, S. K. and Dani, N. A., 2010, p.62-65). The product cordial labeling for some cycle related graphs is discussed in (Vaidya, S. K. and Kanani, K. K., 2010, p.109-116). In the same paper they have investigated product cordial labeling for the shadow graph of cycle Cn.
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1.5 Definition: The tensor product of two graphs G1 and G2 denoted by G1(TP)G2 has vertex set V(G1(TP)G2) = V(G1) × V(G2) and the edge set E(G1(TP)G2) = {(u1, v1)(u2, v2)/u1u2E(G1) and v1v2E(G2)}. We investigate some new results on product cordial labeling in the context of tensor product of graphs. Throughout this work the related graphs are vizulaized through MATGRAPH, which is a toolbox for working with simple graphs in MATLAB. This tool box is available free from http://www.ams.jhu.edu/∼ers/matgraph. 2. Main Results
2.1 Theorem: Pm(T p)Pn is product cordial.
Proof: Let Pm and Pn be two paths of length m − 1 and n − 1 respectively. Let G = Pm(TP)Pn. Denote the vertices of G as uij where 1 ≤ i ≤ m ,1≤ j ≤ n. We note that |V(G)| = mn and |E(G)| = 2(m − 1)(n − 1). We define binary vertex labeling f : V(G) →{0, 1} as follows. 1;ifi + j is even f (u ) = ij 0 ; otherwise.
In view of the above defined labeling pattern the graph G under consideration satisfies the conditions for product cordiality as shown in Table 1. Hence Pm(T p)Pn is product cordial.
2.2 Example: The product cordial labeling for P5(T p)P3 is shown in Figure 1.
2.3 Theorem: Cm(T p)Pn is product cordial for both m and n even.
Proof : Let G = Cm(T p)Pn be the graph obtained by tensor product of Cm and Pn. Denote the vertices of G as uij where 1 ≤ i ≤ m and 1 ≤ j ≤ n. We note that |V(G)| = mn and |E(G)| = 2m(n − 1). We define binary vertex labeling f : V(G) →{0, 1} as follows. 1;ifi + j is even f (u ) = ij 0 ; otherwise.
In view of the above defined labeling pattern the graph G under consideration satisfies the conditions for product cordial labeling as shown in Table 2. Hence Cm(T p)Pn is product cordial for both m and n even.
2.4 Example: The product cordial labeling for C4(T p)P6 is shown in Figure 2.
2.5 Theorem: Cm(T p)Cn is product cordial for m and n even.
Proof: Let Cm and Cn be the cycles with m and n vertices respectively. Let G = Cm(T p)Cn be the graph obtained by tensor product of Cm and Cn where m and n are even. Denote the vertices of G as uij where 1 ≤ i ≤ m and 1 ≤ j ≤ n. We note that |V(G)| = mn and |E(G)| = 2mn. We define binary vertex labeling f : V(G) →{0, 1} as follows. 1;ifi + j is even f (u ) = ij 0 ; otherwise.
In view of the above defined labeling pattern the graph G under consideration satisfies the conditions for product cordial labeling as shown in Table 3. Hence Cm(T p)Pn is product cordial for both m and n even.
2.6 Example: The product cordial labeling for C4(T p)C4 is shown in Figure 3.
2.7 Theorem: The graph obtained by joining two components of Pm(T p)Pn with arbitrary path Pk is product cordial. Proof: Let G = Pm(T p)Pn be the graph obtained by tensor product of Pm and Pn and G be the graph obtained by joining two components of G by a path Pk. Let u1, u2,...,u j and v1, v2,...,v j respectively be the vertices of first and second mn component of G where j = . Let w , w ,...,w be the vertices of path P such that u = w and v = w . We note 2 1 2 k k 1 1 1 k that |V(G)| = mn + k − 2 and |E(G)| = 2mn + k − 1. We define binary vertex labeling f : V(G) →{0, 1} as follows. Case:1 k ≡ 0(mod 2)
f (ui) = 0; 1 ≤ i ≤ j f (vi) = 1; 1 ≤ i ≤ j
84 ISSN 1916-9795 E-ISSN 1916-9809 www.ccsenet.org/jmr Journal of Mathematics Research Vol. 3, No. 3; August 2011 ⎧ ⎪ k ⎨⎪ 0; 1 ≤ i ≤ f (w ) = ⎪ 2 i ⎪ k ⎩ 1; < i ≤ k 2 Case:2 k ≡ 1(mod 2)
f (ui) = 0; 1 ≤ i ≤ j f (vi) = 1; 1 ≤ i ≤ j ⎧ ⎪ k ⎪ 0; 1 ≤ i ≤ ⎨ 2 f (w ) = ⎪ i ⎪ k ⎩⎪ 1; ≤ i ≤ k 2
In view of the above defined labeling pattern f satisfies the conditions for product cordial labeling as shown in Table 4. Thus we prove that the graph obtained by joining two components of Pm(T p)Pn with arbitrary path Pk is product cordial for even m and n.
2.8 Example: The product cordial labeling for the graph obtained by joining two components of P5(T p)P3 by path P4 is shown in Figure 4.
2.9 Theorem: The graph obtained by joining two components of Cm(T p)Pn with arbitrary path Pk is product cordial for m and n even.
Proof: Let Cm be the cycle with m vertices, Pn be the path of length n − 1 and G be the graph obtained by tensor product of Cm and Pn where m and n are even. Let G be the graph obtained by joining two components of G by a path Pk and mn u1, u2,...,u j and v1, v2,...,v j be the vertices of first and second component of G where j = . Let w1, w2,...,wk be 2 the vertices of path Pk with u1 = w1 and v1 = wk. We note that |V(G )| = mn + k − 2 and |E(G )| = 2m(n − 1) + k − 1. We define binary vertex labeling f : V(G) →{0, 1} as follows. Case:1 k ≡ 0(mod 2)
f (ui) = 0; 1 ≤ i ≤ j f (vi) = 1; 1 ≤ i ≤ j ⎧ ⎪ k ⎨⎪ 0; 1 ≤ i ≤ f (w ) = ⎪ 2 i ⎪ k ⎩ 1; < i ≤ k 2 Case:2 k ≡ 1(mod 2)
f (ui) = 0; 1 ≤ i ≤ j f (vi) = 1; 1 ≤ i ≤ j ⎧ ⎪ k ⎪ 0; 1 ≤ i ≤ ⎨ 2 f (w ) = ⎪ i ⎪ k ⎩⎪ 1; ≤ i ≤ k 2
In view of the above defined labeling pattern the graph G satisfies the conditions for product cordial labeling as shown inTable 5. That is, the graph obtained by joining two components of Cm(T p)Pn with arbitrary path Pk is product cordial for even m and n.
2.10 Example: The product cordial labeling for the graph obtained by joining two components of C4(T p)P6 by path P5 is shown in Figure 5.
2.11 Theorem: The graph obtained by joining two components of Cm(T p)Cn with arbitrary path Pk is product cordial for m and n even.
Proof: Let Cm and Cn be the cycle with m and n vertices respectively. Let G be the graph obtained by tensor product of Cm and Cn where m and n are even and G be the graph obtained by joining two components of G by a path Pk. mn Let u , u ,...,u and v , v ,...,v respectively be the vertices of first and second component of G where j = . 1 2 j 1 2 j 2
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Let w1, w2,...,wk be the vertices of path Pk with u1 = w1 and v1 = wk. We note that |V(G )| = mn + k − 2 and |E(G)| = 2mn + k − 1. We define binary vertex labeling f : V(G) →{0, 1} as follows. Case:1 k ≡ 0(mod 2)
f (ui) = 0; 1 ≤ i ≤ j f (vi) = 1; 1 ≤ i ≤ j ⎧ ⎪ k ⎨⎪ 0; 1 ≤ i ≤ f (w ) = ⎪ 2 i ⎪ k ⎩ 1; < i ≤ k 2 Case:2 k ≡ 1(mod 2)
f (ui) = 0; 1 ≤ i ≤ j f (vi) = 1; 1 ≤ i ≤ j ⎧ ⎪ k ⎪ 0; 1 ≤ i ≤ ⎨ 2 f (w ) = ⎪ i ⎪ k ⎩⎪ 1; ≤ i ≤ k 2
In view of the above defined labeling pattern the graph G under consideration satisfies the conditions for product cordial labeling as shown in Table 6. That is, the graph obtained by joining two components of Cm(T p)Cn with arbitrary path Pk is product cordial for even m and n.
2.12 Example: The product cordial labeling for the graph obtained by joining two components of C4(T p)C4 by path P4 is shown in Figure 6. 3. Concluding Remarks As all the graphs are not product cordial graphs it is very interesting to investigate graphs or graph families which admit product cordial labeling. Here we investigate product cordial labeling for some graphs obtained by tensor product of two graphs. To derive similar results for other graph is an open area of research. References Cahit, I. (1987). Cordial Graphs: A weaker version of graceful and harmonious Graphs, Ars Combinatoria, 23, 201-207. Gallian, J. A. (2010). A dynamic survey of graph labeling, The Electronics Journal of Combinatorics, 17(# DS6). [Online] Available: http://www.combinatorics.org/Surveys/ds6.pdf. Sundaram, M. Ponraj, R. & Somsundaram, S. (2004). Product cordial labeling of graphs, Bull. Pure and Applied Sci- ences(Mathematics and Statistics), 23E, 155-163. Vaidya, S. K. & Dani, N. A. (2010). Some new product cordial graphs, Journal of App. Comp. Sci. Math., 8(4), 62-65. [Online] Available: http://www.jacs.usv.ro/getpdf.php?paperid=8 11. Vaidya, S. K. & Kanani, K. K. (2010). Some cycle related product cordial graph, Int. J. of Algorithms, Computing and Math., 3(1), 109-116. [Online] Available: http://eashwarpublications.com/doc/corrected 14.pdf. West, D. B. (2001). Introduction to Graph Theory, Prentice -Hall of India.
Table 1.
Vertex condition Edge condition + = = mn+1 = = − − m and n are odd v f (0) 1 v f (1) 2 e f (0) e f (1) (m 1)(n 1) = = mn = = − − otherwise v f (0) v f (1) 2 e f (0) e f (1) (m 1)(n 1)
Table 2.
Vertex condition Edge condition = = mn = = − m and n are even v f (0) v f (1) 2 e f (0) e f (1) m(n 1)
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Table 3.
Vertex condition Edge condition = = mn = = m and n are even v f (0) v f (1) 2 e f (0) e f (1) mn
Table 4.
Vertex condition Edge condition = = mn+k−2 = + = 2(m−1)(n−1)+k k even v f (0) v f (1) 2 e f (0) e f (1) 1 2 + = = mn+k−1 = = 2(m−1)(n−1)+k−1 k odd v f (0) 1 v f (1) 2 e f (0) e f (1) 2
Table 5.
Vertex condition Edge condition = = mn+k−2 = + = 2m(n−1)+k m and n are even, k even v f (0) v f (1) 2 e f (0) e f (1) 1 2 + = = mn+k−1 = = 2m(n−1)+k−1 m and n are even, k odd v f (0) 1 v f (1) 2 e f (0) e f (1) 2
Table 6.
Vertex condition Edge condition = = mn+k−2 = + = 2mn+k m and n are even, k even v f (0) v f (1) 2 e f (0) e f (1) 1 2 + = = mn+k−1 = = 2mn+k−1 m and n are even, k odd v f (0) 1 v f (1) 2 e f (0) e f (1) 2
Figure 1. The product cordial labeling for P5(T p)P3
Figure 2. The product cordial labeling for C4(T p)P6
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Figure 3. The product cordial labeling for C4(T p)C4
Figure 4. The product cordial labeling for the graph obtained by joining two components of P5(T p)P3 by path P4
Figure 5. The product cordial labeling for the graph obtained by joining two components of C4(T p)P6 by path P5
Figure 6. The product cordial labeling for the graph obtained by joining two components of C4(T p)C4 by path P4
88 ISSN 1916-9795 E-ISSN 1916-9809 ISSN 1923-8444 [Print] Studies in Mathematical Sciences ISSN 1923-8452 [Online] Vol. 3, No. 2, 2011, pp. 11-15 www.cscanada.net DOI: 10.3968/j.sms.1923845220110302.175 www.cscanada.org
E-Cordial Labeling for Cartesian Product of Some Graphs
, S. K. Vaidya1 ∗; N. B. Vyas2
1Department of Mathematics, Saurashtra University, Rajkot-360005, Gujarat, India 2Atmiya Institute of Technology & Science, Rajkot-360003, Gujarat, India ∗Corresponding author. Email: [email protected] Received 10 October, 2011; accepted 27 November, 2011
Abstract We investigate E-cordial labeling for some cartesian product of graphs. We prove that the graphs Kn × P2 and Pn × P2 are E-cordial for n even while Wn × P2 and K1,n × P2 are E-cordial for n odd. Key words E-Cordial labeling; Edge graceful labeling; Cartesian product
S. K. Vaidya, N. B. Vyas (2011). E-Cordial Labeling for Cartesian Product of Some Graphs. Studies in Mathematical Sciences, 3(2), 11-15. Available from: URL: http://www.cscanada.net/index.php/sms/article/view/j.sms.1923845220110302.175 DOI: http://dx.doi.org/10.3968/j.sms.1923845220110302.175
1. INTRODUCTION
We begin with f nite, connected and undirected graph G = (V(G),E(G)) without loops and multiple edges. For standard terminology and notations we refer to West (2001). The brief summary of def nitions and relevant results are given below. Definition 1.1 If the vertices of the graph are assigned values subject to certain condition(s) then it is known as graph labeling. Most of the graph labeling techniques trace their origin to graceful labeling introduced independently by Rosa (1967) and Golomb (1972) which is def ned as follows. Definition 1.2 A function f is called graceful labeling of graph G if f : V(G) → {0, 1, 2,...,q} is injective and the induced function f ∗(e = uv) = | f (u) − f (v)| is bijective. A graph which admits graceful labeling is called a graceful graph. The famous Ringel-Kotzig graceful tree conjecture and illustrious work by Kotzig (1973) brought a tide of labelig problems having graceful theme. Definition 1.3 A graph G is said to be edge-graceful if there exists a bijection f : E(G) → {1, 2,...,|E|} such that the induced mapping f ∗ : V(G) → {0, 1, 2,...,|V| − 1} given by f ∗(x) = f (xy)(mod|V|), xy ∈ E(G). P Definition 1.4 A mapping f : V(G) → {0, 1} is called binary vertex labeling of G and f (v) is called the label of vertex v of G under f . Notations 1.5 For an edge e = uv, the induced edge labeling f ∗ : E(G) → {0, 1} is given by f ∗(e = uv) = | f (u) − f (v)|. Then
v f (i) = number o f vertices o f G having labeli under f ∗ where i = 0 or 1 e f (i) = number o f edges o f G having labeli under f )
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Definition 1.6 A binary vertex labeling of graph G is called a cordial labeling if |v f (0) − v f (1)| ≤ 1 and |e f (0) − e f (1)| ≤ 1. A graph G is called cordial if admits cordial labeling. The concept of cordial labeling was introduced by Cahit (1987). He also investigated several results on this newly introduced concept. Definition 1.7 A function f : E(G) → {0, 1} is called E-cordial labeling of graph G if the induced function ∗ ∗ f : V(G) → {0, 1} def ned by f (v) = f (uv)(mod 2) is such that |v f (0) − v f (1)| ≤ 1 and |e f (0) − uv∈E(G) P e f (1)| ≤ 1. A graph is called E-cordial if admits E-cordial labeling. Yilmaz and Cahit (1997) introduced E-cordial labeling as a weaker version of edge-graceful labeling and having bland of cordial labeling. They proved that the trees with n vertices, the complete graph Kn and cycle Cn are E-cordial if and only if n . 2(mod4) while complete bipartite graph Km,n admits E-cordial labeling if and only if m + n . 2(mod4) Devaraj (2004) has shown that M(m, n) (the mirror graph of Km,n) is E-cordial when m + n is even while the generalized Petersen graph P(n, k) is E-cordial when n is even. Vaidya and Vyas (2011) have proved that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs. , , Definition 1.8 Let G1 = (V1 E1) and G2 = (V2 E2) be two graphs. The cartesian product of G1and G2 which is denoted by G1 × G2 is the graph with vertex set V = V1 × V2 consisting of vertices V = {u = , , , (u1 u2) v = (v1 v2)/u and v are adjacent in G1 × G2whenever u1 = v1 and u2 adjacent to v2 or u1 adjacent to v1 and u2 = v2} In this paper we have investigated some results on E-cordial labeling for cartesian product of some graphs.
2. MAIN RESULTS
Theorem-2.1: Kn × P2 is E-cordial for even n. / , ,..., , Proof: Let G be the graph Kn × P2 where n is even and V(G) = {vi j i = 1 2 n and j = 1 2} be the | | = | | = 2 | | = | | = n(n−1) vertices of graph G We note that V(G) 2n and E(G) n as V(Kn) n and E(Kn) 2 . Def ne f : E(G) → {0, 1} as follows: For 1 ≤ i, k ≤ n , f (vi1 vk1)= 0;
, f (vi2 vk2) = 1;
1; i ≡ 0 (mod2) f (vi , vi ) = 1 2 ( 0; otherwise.
In view of the above def ned labeling pattern f satisf es the conditions for E-cordial labeling as shown in Table 1. That is, Kn × P2 is E-cordial for even n.
Table 1
vertex condition edge condition = = = = n2 n v f (0) v f (1) n e f (0) e f (1) 2
Illustration 2.2: The E-cordial labeling for K4 × P2 is shown in Figure 1.
12 S. K. Vaidya and N. B. Vyas/Studies in Mathematical Sciences Vol.3 No.2, 2011
Figure 1
Theorem-2.3: Wn × P2 is E-cordial for odd n. / , ,..., , Proof: Let G be the graph Wn × P2 where n is odd and V(G) = {vi j i = 1 2 n + 1 and j = 1 2} be the vertices of graph G. We note that |V(G)| = 2(n + 1) and |E(G)| = 5n + 1as |V(Wn)| = n + 1 and |E(Wn)| = 2n. Def ne f : E(G) → {0, 1} as follows: For 1 ≤ i, k ≤ n + 1 , , f (vi1 vk1) = 0; f (vi2 vk2) = 1; , f (vi1 vi2) = 1; i ≡ 0 (mod2) = 0; otherwise. In view of the above def ned labeling pattern f satisf es conditions for E-cordial labeling as shown in Table 2. That is, Wn × P2 is E-cordial for odd n. Table 2
vertex condition edge condition = = + = = 5n+1 n v f (0) v f (1) n 1 e f (0) e f (1) 2
Illustration 2.4: The E-cordial labeling for W3 × P2 is shown in Figure 2.
Figure 2
Theorem-2.5: Ln = Pn × P2(also known as ladder graph) is E-cordial for even n. / , ,..., , Proof: Let G be the graph Pn × P2 where n is even and V(G) = {vi j i = 1 2 n and j = 1 2} be the vertices ofG. We note that |V(G)| = 2n and |E(G)| = 3n − 2. Def ne f : E(G) → {0, 1} as follows:
13 S. K. Vaidya and N. B. Vyas/Studies in Mathematical Sciences Vol.3 No.2, 2011
For 1 ≤ i, k ≤ n , f (vi1 vk1) = 0; , f (vi2 vk2) = 1;
1; i ≡ 0 (mod2) f (vi , vi ) = 1 2 ( 0; otherwise. In view of the above def ned labeling pattern f satisf es conditions for E-cordial labeling as shown in Table 3. That is, Pn × P2 is E-cordial for even n. Table 3
vertex condition edge condition = = = = 3n−2 n v f (0) v f (1) n e f (0) e f (1) 2
Illustration 2.6: The E-cordial labeling for P4 × P2 is shown in Figure 3.
Figure 3
Theorem-2.7: Bn = K1,n × P2(also known as book graph) is E-cordial for odd n. / , ,..., , Proof: Let G be the graph K1,n × P2 where n is odd and V(G) = {vi j i = 1 2 n + 1 and j = 1 2} be the vertices of G. We note that |V(G)| = 2(n + 1) and |E(G)| = 3n + 1. Def ne f : E(G) → {0, 1} as follows: For 1 ≤ i, k ≤ n + 1 , f (vi1 vk1) = 0; , f (vi2 vk2) = 1;
1; i ≡ 0 (mod2) f (vi , vi ) = 1 2 ( 0; otherwise. In view of the above def ned labeling pattern f satisf es the conditions for E-cordial labeling as shown in Table 4. That is, K1,n × P2is E-cordial for odd n. Illustration 2.8: The E-cordial labeling for K1,3 × P2is shown in Figure 4.
14 S. K. Vaidya and N. B. Vyas/Studies in Mathematical Sciences Vol.3 No.2, 2011
Table 4
vertex condition edge condition = = = = 3n+1 n v f (0) v f (1) n e f (0) e f (1) 2
Figure 4
3. CONCLUDING REMARKS
Here we investigate E-cordial labeling for cartesian product of some graphs. Similar results can be derived for other graph families and in the context of different graph labeling problems is an open area of research.
REFERENCES
[1] Cahit I. (1987). Cordial Graphs: A Weaker Version of Graceful and Harmonious Graphs. Ars Combi- natoria, 23, 201-207. [2] Devraj J. (2004). On Edge-Cordial Graphs. Graph Theory Notes of New York, XLVII, 14-18. [3] Gallian, J. A. (2010). A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatorics, 17(#DS6). Retrieved from: http://www.combinatorics.org/Surveys/ds6.pdf [4] Golomb S. W. (1972). How to Number a Graph. In Graph theory and Computing, R. C. Read, ed., New York: Academic Press. pp.23-27. [5] Kotzig A. (1973). On Certain Vertex Valuations of Finite Graphs. Util. Math. 4, 67-73. [6] Rosa A. (1967). On Certain Valuations of the Vertices of a Graph. Theory of graphs (Internat. Sympo- sium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris, 349-355. [7] Vaidya S. K. & Vyas, N. B. (2011). E-Cordial Labeling of Some Mirror Graphs, International Journal of Contemporary Advanced Mathematics, 2(1), 22-27. Retrieved from: http://www.ccsenet. org/journal/index.php/jmr/article/view/9863/8116 [8] Yilmaz R. & Cahit, I. (1997). E-Cordial Graphs, Ars. Combinatoria, 46 , 251-266. [9] West D. B. (2001), Introduction to Graph Theory, Prentice-Hall of India.
15
S K Vaidya & N B Vyas
E-Cordial Labeling of Some Mirror Graphs
S. K. Vaidya [email protected] Department of Mathematics, Saurashtra University, RAJKOT – 360005, Gujarat (India).
N. B. Vyas [email protected] Department of Mathematics, Atmiya Institute of Technology & Science, RAJKOT – 360003, Gujarat (India).
Abstract
Let G be a bipartite graph with a partite sets V1 and V2 and G′ be the copy of G with corresponding partite sets V1′ and V2′ . The mirror graph M(G) of G is obtained from G and G′ by joining each vertex of V1 to its corresponding vertex in V2 by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
Keywords: E-Cordial labeling, Edge graceful labeling, Mirror graphs.
AMS Subject Classification Number(2010): 05C78
1. INTRODUCTION We begin with finite, connected and undirected graph GVGEG= ( ( ), ( )) without loops and multiple edges. For standard terminology and notations we refer to West[1]. The brief summary of definitions and relevant results are given below.
Definition 1.1 If the vertices of the graph are assigned values subject to certain condition(s) then it is known as graph labeling.
Most of the graph labeling techniques trace their origin to graceful labeling introduced independently by Rosa[3] and Golomb[4] which is defined as follows.
Definition 1.2 A function f is called graceful labeling of graph G if f: V ( G )→ {0,1,2, … , q } is injective and the induced function f* ( e= uv ) = | f ( u ) − f ( v ) | is bijective. A graph which admits graceful labeling is called a graceful graph.
The famous Ringel-Kotzig graceful tree conjecture and illustrious work by Kotzig[5] brought a tide of labelig problems having graceful theme.
Definition 1.3 A graph G is said to be edge-graceful if there exists a bijection f: E ( G )→ {1,2, … ,| E |} such that the induced mapping f* : V ( G )→ {0,1,2, … ,| V | − 1} given by f* ( x )= ∑ f ( xy )( mod | V |) , xy∈ E() G .
Definition 1.4 A mapping f: V ( G )→ {0,1} is called binary vertex labeling of G and f( v ) is called the label of vertex v of G under f .
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S K Vaidya & N B Vyas
Notations 1.5 For an edge e= uv , the induced edge labeling f* : E ( G )→ {0,1} is given by * f( e= uv ) = | f ( u ) − f ( v ) | . Let vf(0) , v f ( 1) be the number of vertices of G having labels 0 and 1 respectively under f and let ef(0) , e f ( 1) be the number of edges of G having labels 0 and 1 respectively under f * .
Definition 1.6
A binary vertex labeling of graph G is called a cordial labeling if |vf (0)− v f (1) | ≤ 1 and
|ef (0)− e f (1) | ≤ 1 . A graph G is called cordial if admits cordial labeling.
The concept of cordial labeling was introduced by Cahit[6]. He also investigated several results on this newly introduced concept.
Definition 1.7 Let G be a graph with vertex set VG( ) and edge set EG( ) and let f: E ( G )→ {0,1} . Define on VG( ) by f() v=∑ {() f uv* uv ∈ E ()}( G mod 2) . The function f is called an E-cordial labeling of G if |vf (0)− v f (1) | ≤ 1 and |ef (0)− e f (1) | ≤ 1 . A graph is called E-cordial if admits E-cordial labeling .
In 1997 Yilmaz and Cahit[7] introduced E-cordial labeling as a weaker version of edge-graceful labeling and having flavour of cordial labeling. They proved that the trees with n vertices, K n , Cn are
E-cordial if and only if n≡/ 2( mod 4) while Km, n admits E-cordial labeling if and only if m+ n ≡/ 2( mod 4) .
Definition 1.8
For a bipartite graph G with partite sets V1 and V2 . Let G′ be the copy of G and V1′ and V2′ be the copies of V1 and V2 . The mirror graph MG( ) of G is obtained from G and G′ by joining each ′ vertex of V2 to its corresponding vertex in V2 by an edge.
Lee and Liu[8] have introduced mirror graph during the discussion of k − graceful labeling. Devaraj[9] has shown that M( m, n ) , the mirror graph of Km, n is E-cordial when m + n is even while the generalized Petersen graph P( n, k ) is E-cordial when n is even.
In the following section we have investigated some new results on E-cordial labeling for some mirror graphs.
2. Main Results
Theorem 2.1 Mirror graph of even cycle Cn is E-cordial.
Proof: Let v1,,, v 2 … vn be the vertices and e1,,, e 2 … en be the edges of cycle Cn where n is even andGC= n . Let V1={,,,} v 2 v 4 … vn and V2={,,,} v 1 v 3 … vn− 1 be the partite sets of Cn . Let G′ be the copy of G and V1′′′′={,,,} v 2 v 4 … vn and V′′′′2={,,,} v 1 v 3 … vn− 1 be the copies of V1 and V2 respectively.
Let e1′′′,,, e 2 … en be the edges of G′. The mirror graph MG( ) of G is obtained from G and G′ by *** joining each vertex of V2 to its corresponding vertex in V2′ by additional edges e1,,, e 2 … en . 2 n We note that V( M( G)) = 2n and |E ( M ( G )) |= 2 n + . Let f: E ( M ( G ))→ {0,1} as follows: 2 For 1≤i ≤ n :
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S K Vaidya & N B Vyas
f( e )= 1; i ≡ 0,1(mod4). i = 0;otherwise . f( e′ )= 1; i ≡ 1,2(mod4). i = 0;otherwise .
n or 1F ≤j < : 2 f( e* )= 1; j ≡ 1(mod 2). j = 0;otherwise . n For j = : 2 * f( ej )= 0. In view of the above defined labeling pattern f satisfies conditions for E-cordial labeling as shown in
Table 1. That is, the mirror graph of even cycle C n is E-cordial.
Table 1
Illustration 2.2: The E-cordial labeling for the mirror graph of cycle C 6 is shown in Figure 1.
Figure 1
Theorem 2.3: Mirror graph of path Pn is E-cordial for even n .
Proof: Let v1,,, v 2 … vn be the vertices and e1,,, e 2… en− 1 be the edges of path Pn where n is even and GP= n . Pn is a bipartite graph. Let V1={,,,} v 2 v 4 … vn and V2={,,,} v 1 v 3 … vn− 1 be the bipartition of
Pn . Let G′ be a copy of G and V1′′′′={,,,} v 2 v 4 … vn and V2′′′′={,,,} v 1 v 3 … vn− 1 be the copies of V1 and V2 .
Let e1′′′,,, e 2… en− 1 be the edges of G′.The mirror graph MG( ) of G is obtained from G and G′ by
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*** joining each vertex of V2 to its corresponding vertex in V2′ by additional edges e1,,, e 2 … en . 2 n We note that |V ( M ( G )) |= 2 n and|E ( M ( G )) |= 2( n − 1) + . Let f: E ( M ( G ))→ {0,1} as follows: 2 For 1≤i < n − 1: f( e )= 1; i ≡ 0,1(mod4). i = 0;otherwise . For i= n −1 :
f( ei )= 1. For 1≤i ≤ n − 1: f( e′ )= 1; i ≡ 0,3( mod 4). i = 0;otherwise . n or 1F ≤j ≤ : 2 f( e* )= 1; j ≡ 0(mod2). j = 0;otherwise . In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 2. That is, the mirror graph of path Pn is E-cordial for even n .
Table 2
Illustration 2.4: The E-cordial labeling for mirror graph of path P8 is shown in Figure 2.
Figure 2.
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S K Vaidya & N B Vyas
Theorem 2.5: Mirror graph of hypercube Qk is E-cordial.
Proof: k Let GQ= k be a hypercube with n vertices where n = 2 . Let V1 and V2 be the bipartition of Qk and ′ ′ G′ be a copy of G with V1 and V2 be the copies of V1 and V2 respectively. Let e1,,, e 2 … em be the nk edges of graph G and e′′′,,, e… e be the edges of graph G′ where m = . The mirror graph MG() 1 2 m 2 of G is obtained from G and G′ by joining each vertex of V2 to its corresponding vertex in V2′ by
*** n(2 k + 1) additional edges e1,,, e 2 … en then |V ( M ( G )) |= 2 n and |EMG ( ( ) |= . 2 2 Define f: E ( M ( G ))→ {0,1} as follows:
Case:1 k≡ 0( mod 2)
Let Vi={,,,} v i1 v i 2 … v n and Vi′′′′={,,,} v i1 v i 2 … v n where i =1,2. All the edges incident to the vertices i i 2 2 v1 j and v2′ j where j≡1( mod 2) are assigned the label 0 while the edges incident to the vertices v1 j and v2′ j where j≡ 0( mod 2) are assigned label 1. n For 1≤j ≤ : 2 f( e* )= 1; j ≡ 0(mod2). j = 0;otherwise . Case:2 k≡1( mod 2)
For 1≤i ≤ n :
f( ei )= 1. For 1≤i ≤ n :
f( ei′ )= 0. n or 1F ≤j ≤ : 2 f( e* )= 1; j ≡ 0(mod2). j = 0;otherwise .
In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 3.
TABLE 3
That is, the mirror graph of hypercube Qk is E-cordial.
Illustration:2.6
The E-cordial labeling for mirror graph of hypercube Q3 is shown in Figure 3.
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S K Vaidya & N B Vyas
FIGURE 3
3. CONCLUDING REMARKS Here we investigate E-cordial labeling for some mirror graphs. To investigate similar results for other graph families and in the context of different graph labeling problems is an open area of research.
REFERENCES [1] D B West. Introduction To Graph Theory, Prentice-Hall of India, 2001.
[2] J A Gallian. A dynamic survey of graph labeling, The Electronics Journal of Combinatorics,17, #DS 6 , 2010. Available: http://www.combinatorics.org/Surveys/ds6.pdf
[3] A. Rosa, On certain valuations of the vertices of a graph, Theory of graphs (Internat. Symposium, Rome, July 1966 ), Gordon and Breach, N. Y. and Dunod Paris (1967), 349-355.
[4] S. W. Golomb, How to number a graph, in Graph theory and Computing, R. C. Read, ed., Academic Press, New York (1972), 23-37.
[5] A. Kotzig, On certain vertex valuations of finite graphs, Util. Math., 4(1973), 67-73.
[6] I Cahit, Cordial Graphs: A weaker version of graceful and harmonious Graphs, Ars Combinatoria, 23(1987), 201-207.
[7] R. Yilmaz and I. Cahit, E-cordial graphs, Ars. Combin., 46(1997),251-266.
[8] S. M. Lee and A. Liu, A construction of k-graceful graphs from complete bipartite graphs, SEA Bull. Math., 12(1988), 23-30.
[9] J. Devaraj, On edge-cordial graphs, Graph Theory Notes of New York, XLVII(2004), 14-18.
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Annals of Pure and Applied Mathematics Annals of Vol. 2, No. 1, 2012, 33-39 ISSN: 2279-087X (P), 2279-0888(online) Published on 11 December 2012 www.researchmathsci.org
Antimagic Labeling in the Context of Switching of a Vertex
S. K. Vaidya1 and N. B. Vyas2
1Department of Mathematics, Saurashtra University, Rajkot – 360005, Gujarat, INDIA. E-mail: [email protected]
2Department of Mathematics, Atmiya Institute of Technology & Science, Rajkot – 360005, Gujarat, INDIA. E-mail: [email protected]
Received 30 November 2012; accepted 10 December 2012
Abstract. A graph with q edges is called antimagic if its edges can be labeled with 1, 2,…,q such that the sums of the labels of the edges incident to each vertex are distinct. Here we prove that the graphs obtained by switching of a pendant vertex in path Pn, switching of a vertex in cycle Cn, switching of a rim vertex in wheel Wn, switching of an apex vertex in helm Hn and switching of a vertex of degree 2 in fan fn admit antimagic labeling.
AMS Mathematics Subject Classification (2010): 05C78, 05C38
Keywords: Switching of a vertex, Antimagic labeling, Antimagic graphs.
1. Introduction We begin with a finite, connected and undirected graph G = (V(G),E(G)) without loops and multiple edges. Throughout this paper |V(G)| and |E(G)| respectively denote the number of vertices and number of edges in G. For any undefined notation and terminology we rely upon Gross and Yellen[3]. A brief summary of definitions and existing results is provided in order to maintain the compactness.
Definition 1.1. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (or edges) then the labeling is called a vertex labeling (or an edge labeling).
According to Beineke and Hegde[1] labeling of discrete structure is a frontier between graph theory and theory of numbers. For an extensive survey of graph labeling as well as bibliographic references there in we refer to Gallian[2].
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S. K. Vaidya and N. B. Vyas Definition 1.2. A graph with q edges is called antimagic if its edges can be labeled with 1, 2,…,q such that the sums of the labels of the edges incident to each vertex are distinct.
The concept of antimagic graph was introduced by Hartsfield and Ringel[4]. They showed that paths Pn (n ≥ 3), cycles, wheels, and complete graphs Kn (n ≥ 3) are antimagic. They conjectured that (i) all trees except K2 are antimagic, (ii) all connected graphs except K2 are antimagic. These conjectures are unsettled till today.
Definition 1.3. A vertex switching Gv of a graph G is the graph obtained by taking a vertex v of G, removing all the edges to v and adding edges joining v to every other vertex which are not adjacent to v in G.
Definition 1.4. The wheel graph Wn is defined to be the join K1+Cn. The vertex corresponding to K1 is known as apex vertex and vertices corresponding to cycle are known as rim vertices while the edges corresponding to Cn are known as rim edges. We continue to recognize apex of wheel as the apex of the respective graphs obtained from wheel.
Definition 1.5. The helm Hn is the graph obtained from a wheel Wn by attaching a pendant edge to each rim vertex.
Definition 1.6. A fan graph fn is obtained by Pn+K1.
In the following section we will investigate some new results on Antimagic labeling of graphs.
2. Main Results
Theorem 2.1. Switching of a pendant vertex in a path Pn is antimagic. Proof: Let v1,v2,…,vn be the vertices of Pn and Gv denotes the graph obtained by switching of a pendant vertex v of G = Pn. Without loss of generality let the switched vertex be v1. We note that |(VG )| = n and |(EG )| = 2 n− 4. We define v1 v1 fEG:→− 1,2,...,2 n 4 as follows: ()v1 {}
For 21:≤≤in −
f ()1vvii+1 =− i ; For 3 ≤≤in:
f ()vv1 i =+− n i 4, Above defined edge labeling function will generate all distinct vertex labels as per the definition of antimagic labeling. Hence the graph obtained by switching of a pendant vertex in a path Pn is antimagic.
34
Antimagic Labeling in the Context of Switching of a Vertex
Illustration 2.2. In Figure 1 the graph obtained by switching of a vertex v1 in path P5 and its antimagic labeling is shown.
Figure 1
Theorem 2.3. Switching of a vertex in cycle Cn is antimagic. Proof. Let v1,v2, …,vn be the successive vertices of Cn and Gv denotes graph obtained by switching of vertex v of G= Cn. Without loss of generality let the switched vertex be v1. We note that VG= n and EG= 25 n− . We define ( v1 ) ( v1 ) fEG:→− 1,2,...,2 n 5 as follows: ()v1 {} For 3 ≤≤in:
fvv(1 i )=− 2( i 2) ;
f ()25vvii−1 =− i ,
Above defined edge labeling function will generate all distinct vertex labels as per the definition of an antimagic labeling. Hence the graph obtained by switching of a vertex in a cycle Cn is antimagic.
Illustration 2.4. In Figure 2 the graph obtained by switching of a vertex v1 in cycle C7 and its antimagic labeling is shown.
Figure 2 Theorem 2.5. Switching of a rim vertex in a wheel Wn is antimagic. Proof. Let v as the apex vertex and v1,v2, … ,vn be the rim vertices of wheel Wn. Let
G denotes graph obtained by switching of a rim vertex v1 of G = Wn. We note that v1
35
S. K. Vaidya and N. B. Vyas
VG=+ n 1 and EG= 36 n− . We define fEG:→− 1,2,...,3 n 6 ()v1 ()v1 ( v1 ) {} as follows. For 21≤≤in −:
f ()1vvii+1 =− i ; For 2 ≤≤in:
f ()2vvi =−− n i 1; For 31≤≤in −:
f ()2vv1 i =+− n i 5,
Above defined edge labeling function will generate all distinct vertex labels as per the definition of an antimagic labeling. Hence the graph obtained by switching of a rim vertex in a wheel Wn is antimagic.
Illustration 2.6. In Figure 3 the graph obtained by switching of a vertex v1 in wheel W8 and its antimagic labeling is shown.
Figure 3
Theorem 2.7. Switching of an apex vertex in helm Hn is antimagic. Proof. Let Hn be a helm with v as the apex vertex, v1,v2,…,vn be the vertices of cycle and u1,u2,…,un be the pendant vertices. Let Gv denotes graph obtained by switching of an apex vertex v of G=Hn. We note that VG( v ) = 21 n+ and EG( v ) = 3 n. We define f :EG()v → { 1,2,...,3 n} as follows.
Case 1: n ≡ 0 ( mod 3 ); n ≠ 3
fvu()21 = ;
fvu()111 = ;
fvv()312 = ; For 2 ≤≤in:
36
Antimagic Labeling in the Context of Switching of a Vertex
f ()32vui =− i ;
f ()31vuii =− i ;
f ()3vvii+1 = i where ()vvn+11=
The case when n = 3 is to be dealt separately and the graph is labeled as shown in Figure 4.
Figure 4
Case 2: n ≡ 1, 2 ( mod 3 ) For 1≤≤in:
f ()32vui =− i ;
f ()31vuii =− i ;
f ()3vvii+1 = i where ()vvn+11=
Above defined edge labeling function will generate all distinct vertex labels as per the definition of an antimagic labeling. Hence the graph obtained by switching of an apex vertex in a helm Hn is antimagic.
Figure 5
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S. K. Vaidya and N. B. Vyas
Illustration 2.8. In Figure 5 the graph obtained by switching of an apex vertex v in helm H5 and antimagic labeling is shown.
Theorem 2.9. Switching of a vertex having degree 2 in fan fn is antimagic.
Proof. Let v as the apex vertex and v1, v2, … , vn be the vertices of fan fn. Let G v1 denotes graph obtained by switching of a vertex v1 having degree 2 of G = fn. We note that VG=+ n 1 and EG= 35 n− . ()v1 ( v1 ) We define fEG:→− 1,2,...,3 n 5 as follows. ()v1 {} For 2 ≤≤in:
f ()vvi =− i 1; For 21≤≤in −:
f ()vvii+1 =+− n i 2; For 3 ≤≤in:
f ()2vv1 i =+− n i 5;
Above defined edge labeling function will generate all distinct vertex labels as per the definition of an antimagic labeling. Hence the graph obtained by switching of a vertex having degree 2 in fan fn is antimagic.
Illustration 2.10. In Figure 6 the graph obtained by switching of a vertex v1 having degree 2 in fan f5 and its antimagic labeling is shown.
Figure 6 3. Concluding Remarks The investigations reported here is an effort to relate graph operations and antimagic labeling. To investigate some characterization(s) or sufficient condition(s) for any graph to be antimagic is an open area of research.
38
Antimagic Labeling in the Context of Switching of a Vertex REFERENCES
1. L. W. Beineke and S. M. Hegde, Strongly Multiplicative graphs, Discuss. Math. Graph Theory, vol. 21, pp. 63-75, 2001. 2. J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronics Journal of Combinatorics, vol. 18 (#DS6), 2011. Available: http://www.combinatorics.org/Surveys/ds6.pdf 3. J. Gross and J. Yellen, Graph Theory and its Applications, CRC Press, 1999. 4. N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston, 1990.
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International Journal of Information Science and Computer Mathematics Volume 5, Number 1, 2012, Pages 9-18 Available online at http://www.pphmj.com/journals/ijiscm.htm Published by Pushpa Publishing House, Allahabad, INDIA
FURTHER RESULTS ON E-CORDIAL LABELING
S. K. Vaidya∗ and N. B. Vyas Department of Mathematics Saurashtra University Rajkot-360 005, Gujarat, India e-mail: [email protected] Atmiya Institute of Technology and Science Rajkot-360 005, Gujarat, India e-mail: [email protected]
Abstract
Here we investigate E-cordial labeling of some graphs. We prove that
the Möbius ladder Mn, middle graph of Cn and crown n :KC 1
admit E-cordial labeling for even n while bistar B ,nn and its square 2 graph B, nn admit E-cordial labeling for odd n.
1. Introduction
Throughout this work, by = (), ()()GEGVG we mean a finite, connected and undirected graph without loops and multiple edges. We will give brief summary of definitions and existing results.
Definition 1.1. If the vertices of the graph are assigned values subject to certain condition(s), then it is known as graph labeling. © HousePublishingPushpa2012 2010 Mathematics Subject Classification: 05C78. Keywords and phrases: E-cordial labeling, Möbius ladder, middle graph, crown, bistar. ∗Corresponding author Communicated by Kewen Zhao Received March 15, 2012 S. K. Vaidya and N. B. Vyas 10 S. K. Vaidya and N. B. Vyas
Most of the graph labeling techniques trace their origin to graceful labeling introduced independently by Rosa [7] and Golomb [3] defined as follows.
Definition 1.2. A function ( ) → { ...,,1,0: (GEGVf ) } is called graceful labeling of graph G if f is injective and the induced function f ∗ ()→ {}...,,2,1: ()GEGE defined by ∗()()()−== vfufuvef is bijective. A graph which admits graceful labeling is called a graceful graph. The famous Ringel-Kotzig conjecture [6] and many illustrious works on it brought a tide of labeling problems with graceful theme.
Definition 1.3. A graph G is said to be edge-graceful if there exists a bijection f ()→ {}...,,2,1: (GEGE ) such that the induced function f ∗ : ()→ {} ()GVGV − 1...,,2,1,0 defined by ∗()= ∑ ( )() (GVxyfxf ) ,mod taken over all edges xy is a bijection.
The notion of edge gracefulness was introduced by Lo [5].
Definition 1.4. A mapping (GVf ) → { 1,0: } is called binary vertex labeling of G and (vf ) is called the label of vertex v of G under f.
Notation. For an edge = uve , the induced edge labeling ∗ :()GEf → ∗ {}1,0 is given by ()()()−== vfufuvef , then f (iu ) = the number of vertices of G having label i under f and let f (ie ) = the number of edges of G having label i under f ∗ for i = .1,0
Definition 1.5. A binary vertex labeling of graph G is called a cordial labeling if ()− vv ff () ≤ 110 and ( ) − ee ff ( ) ≤ .110 A graph G is called cordial if admits cordial labeling.
The concept of cordial labeling was introduced by Cahit [1]. He also investigated several results on this newly defined concept.
Definition 1.6. Let G be a graph with vertex set (GV ) and edge set Further Results on E-cordial Labeling 11
(GE ) and let ()GEf → { }.1,0: Define f ∗ on (GV ) by ∗ ()vf = ∑ ()∈| (){}(GEuvuvf ).2mod The function f is called an E-cordial labeling of G if ()vv ff () ≤− 110 and ( ) − ee ff ( ) ≤ .110 A graph is called E-cordial if admits E-cordial labeling. In 1997, Yilmaz and Cahit [12] introduced E-cordial labeling as a weaker version of edge-graceful labeling having the blend of cordial labeling. They proved that the trees with n vertices, Kn, Cn are E-cordial if and only if n ≡/ ()4mod2 while K , nm admits E-cordial labeling if and only if + nm ≡/ ().4mod2
Vaidya and Bijukumar [8] proved that the graphs obtained by duplication of an arbitrary vertex as well as an arbitrary edge in cycle Cn admit E- cordial labeling. In addition to this, they also derived that the joint sum of two copies of cycle Cn , the split graph of even cycle Cn and the shadow graph of path Pn for even n are E-cordial graphs. The same authors in [9] proved that the middle graph, total graph and split graph of Pn and the composition of Pn with P2 admit E-cordial labeling.
Vaidya and Vyas [10] proved that the mirror graphs of even cycle Cn , even path Pn and hypercube Qk are E-cordial graphs. The same authors in
[11] proved that n × PK 2 and n × PP 2 are E-cordial graphs for even n while
n × PW 2 and n × PK 2,1 are E-cordial graphs for odd n.
Definition 1.7. The middle graph (GM ) of a graph G is the graph whose vertex set is ()∪ (GEGV ) and in which two vertices are adjacent if and only if either they are adjacent edges of G or one is a vertex of G and the other is an edge incident on it.
Definition 1.8. For a simple graph G, the square of graph G is denoted by G2 and defined as the graph obtained with the same vertex set as of G and two vertices are adjacent in G2 if they are at a distance 1 or 2 apart in G. S. K. Vaidya and N. B. Vyas 12 S. K. Vaidya and N. B. Vyas
In the following section, we investigate some new results on E-cordial labeling of graphs. The definition of special graph families can be found in Gallian [2] while for standard graph theoretic notations and terminology we refer to Harary [4].
2. Main Results
Theorem 2.1. Möbius ladder Mn is E-cordial for even n.
Proof. Let =MG n be a Möbius ladder graph for even n. We note that () = 2nGV and () = nGE .3 We define (GEf ) → { 1,0: } as follows:
( 1vvf n′ ) = ,1
( nvvf 1′ ) = .1
For <≤ ni :1
⎧ i ≡ ( ),2mod0,1 ()vvf ii +1 = ⎨ ⎩ otherwise,,0
⎧ i ≡ ( ),2mod0,1 ()vvf ii ′′ +1 = ⎨ ⎩ otherwise.,0
For ≤≤ ni :1
⎧ i ≡ ( ),2mod0,1 ()vvf ii ′ = ⎨ ⎩ otherwise.,0
In view of the above defined labeling pattern f satisfies the conditions for
E-cordial labeling as shown in Table 1. Hence Mn is E-cordial for even n.
Table 1 vertex condition edge condition 3n even n () = (10 ) = nvv ()()ee 10 == ff ff 2 Further Results on E-cordial Labeling 13
Illustration 2.2. E-cordial labeling of M 6 is shown in Figure 1.
Figure 1
Theorem 2.3. (CM n ) admits E-cordial labeling for even n.
Proof. Let = (CMG n ) be a middle graph of cycle Cn for even n. We note that ( ) = 2nGV and ( ) = nGE .3 We define (GEf ){}→ 1,0: as follows: For <≤ ni :1 ⎧ i ≡ ( ),2mod0,0 ()vvf ii +1 = ⎨ ⎩ otherwise,,1
()nvvf 1 = ,0
()n′ vvf 1 = ,1
()′ ii = ≤ ≤ nivvf ,1;0
()′ ii +1 = ≤ < nivvf .1;1 In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 2. Hence (CM n ) is E-cordial for even n. S. K. Vaidya and N. B. Vyas 14 S. K. Vaidya and N. B. Vyas
Table 2 vertex condition edge condition 3n even n () = (10) = nvv ()()ee 10 == ff ff 2
Illustration 2.4. E-cordial labeling of (CM 6 ) is shown in Figure 2.
Figure 2
Theorem 2.5. The crown n :KC 1 is E-cordial for even n.
Proof. Let 21 ...,,, vvv n be the vertices of Cn , where n is even, while
21 ...,,, uuu n be the newly added pendant vertices in order to construct G and denote n : 1 = GKC . We note that ( ) = 2nGV and ( ) = nGE .2 We define (){}GEf → 1,0: as follows:
For ni −≤≤ :11
⎧ i ≡ ( ),2mod1,1 ()()vf u vf v iiii +1 == ⎨ ⎩ otherwise,,0
()vf u = (vf nnn v 1) = .0 Further Results on E-cordial Labeling 15
In view of the above defined labeling pattern f satisfies the conditions for
E-cordial labeling as shown in Table 3. Hence crown n : KC 1 is E-cordial for even n. Table 3 vertex condition edge condition
even n ( ) = ff (10) = nvv ( ) = ff (10) = nee
Illustration 2.6. E-cordial labeling of crown : KC 16 is shown in Figure 3.
Figure 3
Theorem 2.7. The bistar B , nn is E-cordial for odd n.
Proof. Let = BG , nn for odd n with vertex set { ii ≤ ≤ nivuvu },1,,,, where ui , vi are pendant vertices. Then ( ) = nGV + 22 and ()GE = n + .12 We define ()GEf → { 1,0: } as follows:
For ≤≤ ni :1 (uvf ) = ,0
(uuf i ) = ,0
(vvf i ) = .1 S. K. Vaidya and N. B. Vyas 16 S. K. Vaidya and N. B. Vyas
In view of the above defined labeling pattern f satisfies the conditions for
E-cordial labeling as shown in Table 4. Hence bistar B , nn is E-cordial for odd n. Table 4 vertex condition edge condition
odd n ()= ff ( ) = nvv + 110 ( ) = ff ( ) + = nee + 1110
Illustration 2.8. E-cordial labeling of B 5,5 is shown in Figure 4.
Figure 4
2 Theorem 2.9. B nn is E-cordial for odd n.
Proof. Consider B , nn for odd n with vertex set { ii ≤ ≤ nivuvu },1,,,, 2 where ui , vi are pendant vertices. Let G be the graph B , nn . Then ()GV n += 22 and ()= nGE + .14 We define (GEf ) → { 1,0: } as follows:
For ≤≤ ni :1 ()vuf = ,1
()vvf i = ,1
()uvf i = ,0 n − 1 ⎪⎧ 1,1 i ≤≤ , () ()vufuuf ii == ⎨ 2 ⎩⎪ otherwise.,0 Further Results on E-cordial Labeling 17
In view of the above defined labeling pattern f satisfies the conditions for 2 E-cordial labeling as shown in Table 5. Hence bistar B , nn is E-cordial for odd n. Table 5 vertex condition edge condition
odd n ()= ff ( ) = nvv + 110 ( ) = ff ( ) + = nee + 12110
2 Illustration 2.10. E-cordial labeling of B 5,5 is shown in Figure 5.
Figure 5
3. Concluding Remarks
Some new families of E-cordial graphs are investigated. To investigate similar results for other graph families and in the context of different graph labeling problems is an open area of research.
References
[1] I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987), 201-207. S. K. Vaidya and N. B. Vyas 18 S. K. Vaidya and N. B. Vyas
[2] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 18 (2011), #DS6. [3] S. W. Golomb, How to number a graph, Graph Theory and Computing, R. C. Read, ed., Academic Press, New York, 1972, pp. 23-37. [4] F. Harary, Graph Theory, Addison Wesley, Reading, Massachusetts, 1969. [5] S. Lo, On edge graceful labelings of graphs, Congr. Numer. 50 (1985), 231-241. [6] G. Ringel, Problem 25, Theory of Graphs and its Applications, Proc. Symposium Smolenice, 1963, Prague, 1964, p. 162. [7] A. Rosa, On certain valuations of the vertices of a graph, Theory of graphs, Internat. Symposium, Rome, July 1966, Gordon and Breach, N. Y. and Dunod, Paris, 1967, pp. 349-355. [8] S. K. Vaidya and Lekha Bijukumar, Some new families of E-cordial graphs, J. Math. Res. 3 (2011), 105-111. [9] S. K. Vaidya and Lekha Bijukumar, Some new results on E-cordial labeling, Int. J. Inform. Sci. Comp. Math. 3(1) (2011), 21-29. [10] S. K. Vaidya and N. B. Vyas, E-cordial labeling of some mirrorgraphs, Int. J. Contemp. Adv. Math. 2 (2011), 22-27. [11] S. K. Vaidya and N. B. Vyas, E-cordial labeling for Cartesian product of some graphs, Stud. Math. Sci. 3 (2011), 11-15.
[12] R. Yilmaz and I. Cahit, E-cordial graphs, Ars Combin. 46 (1997), 251-266. International Journal of Advanced Computer and Mathematical Sciences ISSN 2230-9624. Vol 3, Issue 4, 2012, pp 446-452 http://bipublication.com
E-CORDIAL LABELING IN THE CONTEXT OF SWITCHING OF A VERTEX
S. K. Vyaidya1 and N. B. Vyas 2
1Department of Mathematics, Saurashtra University, Rajkot-360005 (Gujarat) – India. 2Department of Mathematic, Atmiya Institute of Technology & Science, Rajkot – 360005 (Gujarat) – India. *Corresponding author: Email: [email protected], Tel: +91-9825292539
[Received-07/07/2012, Accepted-03/09/2012]
ABSTRACT:
Let G=(V(G),E(G)) be a graph and f: E ( G )→ {0,1} be a binary edge labeling. Define f* : V ( G )→ {0,1} by * = − ≤ f( v )∑ f ( uv )( mod 2) . The function f is called E-cordial labeling of G if |vf (0) v f (1) | 1 and uv∈ E() G − ≤ |ef (0) e f (1) | 1 . In the present work we discuss E-cordial labeling in the context of switching of a vertex in cycle, wheel, helm and closed helm.
Keywords: E-cordial labeling , Cycle, Wheel, Vertex switching
[I] INTRODUCTION condition(s) then it is known as graph We begin with finite, connected and labeling. undirected graph GVGEG= ( ( ), ( )) without Most of the graph labeling techniques trace loops and multiple edges. For standard their origin to graceful labeling introduced terminology and notations we follow independently by Rosa[7] and Golomb[3] Harary[4]. For extensive survey of graph which is defined as follows. labeling as well as bibliographic references 1.2. Definition we refer Gallian[2]. A function f: V ( G )→ {0,1, … ,| E ( G )|} is 1.1. Definition If the vertices of the graph called graceful labeling of graph G if f is are assigned values subject to certain injective and the induced function f* : E ( G )→ {1,2, … ,| E ( G )|} defined by E-CORDIAL LABELING IN THE CONTEXT OF SWITCHING OF A VERTEX
f* ( e= uv ) = | f ( u ) − f ( v ) | is bijective. A 1.7. Definition graph which admits graceful labeling is Let G be a graph with vertex set V(G) and → called a graceful graph. edge set E(G) and let f: E ( G ) {0,1}. The famous Ringel-Kotzig conjecture [6] Define f* on V(G) by and illustrious work on it brought a tide of f* () v=∑ {() f uv* uv ∈ E ()}( G mod 2). The labeling problems with graceful theme. function f is called an E-cordial labeling of 1.3. Definition G if |v (0)− v (1) | ≤ 1 and A graph G is said to be edge-graceful if f f − ≤ there exists a bijection |ef (0) e f (1) | 1. A graph is called E- f:() E G→ {1,2,,|()|} … E G such that the cordial if it admits E-cordial labeling. induced function In 1997 Yilmaz and Cahit[12] have f* : V ( G )→ {0,1, 2, … ,| V ( G ) | − 1} defined introduced E-cordial labeling as a weaker version of edge-graceful labeling having the by f* ( x )= ( f ( xy ))( mod | V ( G ) |) , taken ∑ blend of cordial labeling. They proved that over all edges xy is a bijective. the trees with n vertices, Kn, Cn are E-cordial
The notion of edge gracefulness was if and only if n≡/ 2( mod 4) while Km,n introduced by Lo[5]. admits E-cordial labeling if and only if 1.4. Definition m+ n ≡/ 2( mod 4) . A mapping f: V ( G )→ {0,1} is called Vaidya and Lekha[8] have proved that the binary vertex labeling of G and f(v) is called graphs obtained by duplication of an the label of vertex v of G under f. arbitrary vertex as well as an arbitrary edge 1.5. Notation in cycle Cn admit E-cordial labeling. In For an edge e=uv, the induced edge labeling addition to this they also derived that the * → f: E ( G ) {0,1} is given by joint sum of two copies of cycle Cn, the split * f( e= uv ) = | f ( u ) − f ( v ) | then vf(i) = the graph of even cycle Cn and the shadow number of vertices of G having label i graph of path Pn for even n are E-cordial graphs. The same authors in [9] proved that under f and let ef(i) = the number of edges of G having label i under f* for i=0,1. the middle graph, total graph and split graph 1.6. Definition of Pn and the composition of Pn with P2 A binary vertex labeling of graph G is called admit E-cordial labeling. − ≤ Vaidya and Vyas[10] proved that the mirror a cordial labeling if |vf (0) v f (1) | 1and graphs of even cycle Cn, even path Pn and |e (0)− e (1) | ≤ 1. A graph G is called f f hypercube Qk are E-cordial graphs. The cordial if it admits cordial labeling. same authors in [11] proved that Kn × P2 and The concept of cordial labeling was Pn × P2 are E-cordial graphs for even n introduced by Cahit[1]. He also investigated while Wn × P2 and K1,n × P2 are E-cordial several results on this newly defined graphs for odd n. concept. 1.8. Definition
A vertex switching Gv of a graph G is the graph obtained by taking a vertex v of G,
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removing all the edges incident to v and Case 1: When n is odd adding edges joining v to every other vertex For 3≤i ≤ n − 1: which are not adjacent to v in G. 1,i ≡ 1(mod 2) = f() v1 vi 1.9. Definition 0,otherwise ; The wheel graph W is defined to be the join n Subcase 1: n≡ 3( mod 4) K1+Cn. The vertex corresponding to K1 is For 2≤i ≤ n − 1: known as apex vertex and vertices 1,i ≡ 0( mod 2) corresponding to cycle are known as rim = f() vi v i+1 vertices while the edges corresponding to 0,otherwise ; cycle are known as rim edges. We continue Subcase 2: n≡1( mod 4) to recognize apex of wheel as the apex of For 2≤i ≤ n − 1: respective graphs obtained from wheel. 0,i ≡ 0( mod 2) = 1.10. Definition f() vi v i+1 1,otherwise ; The helm Hn is the graph obtained from a Case 2: When n is even wheel Wn by attaching a pendant edge to Subcase 1: n≡ 0( mod 4) each rim vertex. = 1.11. Definition f( v2 v 3 ) 1;
The closed helm CHn is the graph obtained For 3≤i ≤ n − 1: from a helm Hn by joining each pendant 0,i ≡ 1(mod 2) = f() v1 vi vertex to form a cycle. 1,otherwise ; We continue to recognize terminology used 1,i ≡ 1(mod 2) = in Definition 1.9 for Definitions 1.10 and f() vi v i+1 1.11 also. 0,otherwise ; In the following section we will investigate Subcase 2: n≡ 2( mod 4) some new results on E-cordial labeling of A graph with n vertices is not E-cordial graphs. when n≡ 2( mod 4) as observed by Yilmaz and Cahit [12].
[II] MAIN RESULTS 2.1. Theorem In view of the labeling pattern defined above − ≤ The graph obtained by switching of an f satisfies the condition |vf (0) v f (1) | 1 arbitrary vertex in cycle Cn admits E-cordial and |e (0)− e (1) | ≤ 1 as shown in Table 1. labeling except for n≡ 2( mod 4) . f f Hence Gv admits E-cordial labeling. Proof: Let v1,v2,…,vn be the successive 1 vertices of Cn and Gv denotes graph obtained 2.2. Illustration by switching of vertex v of G. Without loss Consider the graph obtained by switching of of generality let the switched vertex be v1 a vertex v1 in cycle C7. The E-cordial and we initiate the labeling from v1. To labeling is as shown in Figure 1. define f: E ( G )→ {0,1} we consider v1 following cases.
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= f( vi v i+1 ) 0; For 3≤i ≤ n − 1: n +1 = 0, 3„„i f() v1 vi 2 1,otherwise . Subcase 2: n≡1( mod 4) = + ≡ In this case |V ( Wn | n 1 2() mod 4) ≡ Figure 1 equivalently n1( mod 4). This graph is not 2.3. Theorem E-cordial because the graph G with number The graph obtained by switching of a rim of vertices congruent to 2(mod 4) does not vertex in wheel Wn admits E-cordial labeling admits E-cordial labeling as observed by except for n≡1( mod 4) . Yilmaz and Cahit [12]. Proof: Let v as the apex vertex and In view of the labeling pattern defined above − ≤ v1,v2,..,vn be the rim vertices of wheel Wn. f satisfies the condition |vf (0) v f (1) | 1 Let G denotes graph obtained by switching − ≤ v1 and |ef (0) e f (1) | 1 as shown in Table 2. of a rim vertex v1 of G=Wn. We define Hence G admits E-cordial labeling. v1 f: E ( G )→ {0,1} as follows. v1 2.4. Illustration Case 1: When n is even Consider the graph obtained by switching of ≤ ≤ For 2 i n : a rim vertex in wheel W8. The E-cordial 1,i ≡ 0( mod 2) labeling is as shown in Figure 2. = f() vvi 0,otherwise . For 2 ≤i ≤ n : 1,i ≡ 0( mod 2) = f() vvi 0,otherwise . For 2≤i ≤ n − 1: 1,i ≡ 0( mod 2) = f() vi v i+1 0,otherwise . ≤ ≤ − For 3i n 1: n Figure 2 0, 3„„i + 1 2.5. Theorem f() v v = 2 1 i The graph obtained by switching of an apex 1,otherwise . vertex in helm Hn admits E-cordial labeling. Case 2: When n is odd Proof: Let Hn be a helm with v as the apex Subcase 1: n≡ 3( mod 4) vertex, v1,v2,…,vn be the vertices of cycle For 2 ≤i ≤ n : and u1,u2,…,un be the pendant vertices. Let = f( vvi ) 1; Gv denotes graph obtained by switching of For 2≤i ≤ n − 1:
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an apex vertex v of G=Hn. We define of an apex vertex v of G=CHn. We define → → f: E ( Gv ) {0,1} as follows. f: E ( Gv ) {0,1} as follows. For 1≤i ≤ n : For 1≤i ≤ n : = = f( vui ) 1; f( vui ) 1;
f( v v+ )= 0; ( v + = v ) f( v v )= 0; ( v = v ) i i1 n 1 1 i i+1 n + 1 1 = = n f( u u+ ) 0; ( u + u ) 0, 1„„i i i1 n 1 1 = = f() vi u i 2 f( vi u i ) 1. 1,otherwise . In view of the labeling pattern defined above − ≤ In view of the labeling pattern defined above f satisfies the condition |vf (0) v f (1) | 1 f satisfies the condition |v (0)− v (1) | ≤ 1 − ≤ f f and |ef (0) e f (1) | 1 as shown in Table 4. − ≤ and |ef (0) e f (1) | 1 as shown in Table 3. Hence Gv admits E-cordial labeling.
Hence Gv admits E-cordial labeling. 2.8. Illustration 2.6. Illustration Consider the graph obtained by switching of Consider the graph obtained by switching of an apex vertex in closed helm CH5. The E- an apex vertex in Helm H6. The E-cordial cordial labeling is as shown in Figure 4. labeling is as shown in Figure 3.
Figure 4 Figure 3 2.7. Theorem The graph obtained by switching of an apex [III] CONCLUDING REMARKS Here we investigate E-cordial labeling in the vertex in closed helm CHn admits E-cordial labeling. context of switching of a vertex of some graphs. To investigate similar results for Proof: Let v as the apex vertex, v1,v2,…, vn other graph families and in the context of be the vertices of inner cycle and u1, different graph labeling problems is an open u2,…,un be the vertices of outer cycle CHn. area of research. Let Gv denotes graph obtained by switching
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vertex condition edge condition − ≡()()= = n + = = 2n 5 n0( mod 4)vf0 v1 f ef (0) 1 e f (1) 2 2 − ≡()()= = n = + = 2 n 5 n1( mod 4)vf0 v f 1 +1 ef (0) e f (1) 1 2 2 − ≡()()= = n + = = 2 n 5 n3( mod 4)vf 0 +1 v 1 +1 ef (0) 1 e ff ( 1) 2 2
Table 1: vertex and edge conditions corresponding to Theorem 2.1
vertex condition edge condition n 3(n − 2) n ≡0(mod4)v()() 0 −1 = v1 = e (0) = e (1) = f f 2f f 2 3(nn − 2) n ≡2(mod 4) v()() = v1− = e(0)= e (1)01 = f f 2f f 2 n +1 3(n − 2) + 1 ≡3(mod 4)v()() 0 = v1 = (0)+ 1 =ene (1) = 2ff ff 2
Table 2: vertex and edge conditions corresponding to Theorem 2.3
vertex condition edge condition n n≡0( mod2 )v()() 0− 1 = v1 = n e (0) = e (1) = n + f f f f 2 ≡()()= − = = + = + n n1( m od2 )vf 0 v f 1 1 n ef (0) e f (1) 1 n 2
Table 3: vertex and edge conditions corresponding to Theorem 2.5 vertex condition edge condition ≡()()− = = = = n0( mod2 )v 0 1 v 1 n ef (0) e fff (1) 2 n ≡()()= − = = = n1( mod2 )vf 0 v 1 1 n e f (0) e ff (1) 2 n
Table 4: vertex and edge conditions corresponding to Theorem 2.7
REFERENCES [5] Lo S., [1985] On edge graceful labelings of [1] Cahit I., [1987] Cordial Graphs: A weaker graphs, Congr. Numer. 50: 231-241. version of graceful and harmonious Graphs, Ars [6] Ringel G., [1964] Problem 25, Theory of Graphs Combinatoria 23: 201-207. and its Applications, Proc. Symposium [2] Gallian J. A., [2011] A dynamic survey of graph Smolenice, 1963, Prague, 162. labeling, The Electronics Journal of [7] Rosa A., [1967] On certain valuations of the Combinatorics, 18 (#DS6). Available: vertices of a graph, Theory of graphs http://www1.combinatorics.org/Surveys/ds6.pdf (Internat. Symposium, Rome, July 1966 ), [3] Golomb S. W., [1972] How to number a graph, Gordon and Breach, N. Y. and Dunod Paris, in Graph theory and Computing, R. C. Read, ed., 349-355. Academic Press, New York, 23-37. [8] Vaidya S. K. and Lekha Bijukumar, [2011] [4] Harary F., [1969] Graph Theory, Addison Some New Families of E-cordial Graphs, Wesley, Reading, Massachusetts.
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Journal of Mathematics Research, 3: 105-111. doi:10.5539/jmr.v3n4p105 [9] Vaidya S. K. and Lekha Bijukumar, [2011] Some New Results on E-cordial Labeling, Int. Journal of Information Science and Computer Mathematics 3: 21-29. Available: http://www.pphmj.com/abstract/6004.htm [10] Vaidya S. K. and Vyas N. B., [2011] E-cordial Labeling of Some Mirror graphs, Int. Journal of Contemporary Advanced Mathematics 2: 22-27. Available: http://www.cscjournals.org/csc/manuscript/Journ als/IJCM/volume2/Issue1/IJCM-19.pdf [11] Vaidya S. K. and Vyas N. B., [2011] E-cordial Labeling for Cartesian Product of Some Graphs, Studies in Mathematical Sciences 3: 11-15. doi:10.3968/j.sms.1923845220110302.175 [12] Yilmaz R. and Cahit I., [1997] E-cordial graphs, Ars Combin. 46: 251-266.
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INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 9 Some Results on E-cordial Labeling S. K. Vaidya and N. B. Vyas
Abstract—A binary vertex labeling f : E(G) → {0, 1} with with graceful theme. induced labeling f ∗ : V (G) → {0, 1} defined by f ∗(v) = P {f(uv) | uv ∈ E(G)}(mod 2) is called E-cordial labeling of Definition 1.3: A graph G is said to be edge-graceful if there a graph G if the number of vertices labeled 0 and number of vertices labeled 1 differ by at most 1 and the number of edges exists a bijection f : E(G) → {1, 2,...,|E(G)|} such that ∗ labeled 0 and the number of edges labeled 1 differ by at most the induced mapping f : V (G) → {0, 1, 2,...,|V (G)| − 1} 1. A graph which admits E-cordial labeling is called E-cordial given by f ∗(x) = P f(xy))(mod |V |), taken over all edges graph. Here we prove that flower graph F ln, closed helm CHn, xy is a bijective. double triangular snake DTn and gear graph Gn are E-cordial graphs. The notion of edge gracefulness was introduced by Lo[6]. Index Terms—Binary vertex labeling, Cordial labeling, E-cordial labeling, E-cordial graphs. Definition 1.4: A mapping f : V (G) → {0, 1} is called MSC 2010 Codes - 05C78, 05C38. binary vertex labeling of G and f(v) is called the label of vertex v of G under f.
I.INTRODUCTION Notation: For an edge e = uv, the induced edge labeling E begin with finite, connected and undirected graph f ∗ : E(G) → {0, 1} is given by f ∗(e = uv) = |f(u) − f(v)| G = (V (G),E(G)) without loops and multiple edges. W then vf (i) = the number of vertices of G having label i under Throughout this paper |V (G)| and |E(G)| respectively denote f and let ef (i) = the number of edges of G having label i the number of vertices and number of edges in G. For any under f ∗ for i = 0, 1. undefined notation and terminology we rely upon Gross and Yellen [5]. In order to maintain compactness we will provide Definition 1.5: A binary vertex labeling of graph G is called a brief summary of definitions and existing results. cordial labeling if |vf (0)−vf (1)| ≤ 1 and |ef (0)−ef (1)| ≤ 1. A graph G is called cordial if it admits cordial labeling. Definition 1.1: A graph labeling is an assisgnment of integers to the vertices or edges or both subject to certain condition(s). The concept of cordial labeling was introduced by Cahit[2]. If the domain of the mapping is the set of vertices (or edges) He also investigated several results on this newly defined then the labeling is called a vertex labeling (or an edge concept. labeling). Definition 1.6: Let G be a graph with vertex set V (G) Beineke and Hegde[1] describe labeling of discrete and edge set E(G) and let f : E(G) → {0, 1}. Define f ∗ structure as a frontier between graph theory and theory of on V (G) by f ∗(v) = P{f(uv) | uv ∈ E(G)}(mod 2). numbers. For extensive survey of graph labeling as well as The function f is called an E − cordial labeling of G if bibliographic references we refer Gallian[3]. |vf (0) − vf (1)| ≤ 1 and |ef (0) − ef (1)| ≤ 1. A graph is called E − cordial if it admits E − cordial labeling . Most of the graph labeling techniques trace their origin to graceful labeling introduced independently by Rosa[8] and In 1997 Yilmaz and Cahit[13] introduced E-cordial Golomb[4] defined as follows. labeling as a weaker version of edge-graceful labeling and with the blend of cordial labeling. They proved that the Definition 1.2: A function f : V (G) → {0, 1,...,|E(G)|} is trees with n vertices, Kn, Cn are E-cordial if and only if called graceful labeling of graph G if f is injective and the n 6≡ 2(mod 4) while K admits E-cordial labeling if and ∗ m,n induced function f (e = uv) = |f(u) − f(v)| is bijective. only if m + n 6≡ 2(mod 4). A graph which admits graceful labeling is called a graceful graph. Vaidya and Lekha[9] have proved that the graphs obtained by duplication of an arbitrary vertex as well as an arbitrary The famous Ringel-Kotzig conjecture [7] and many edge in cycle Cn admit E-cordial labeling. In addition to this illustrious work on it brought a tide of labeling problems they show that the joint sum of two copies of cycle Cn, the S. K. Vaidya is working as a professor at the Department of Mathematics, split graph of even cycle Cn and the shadow graph of path Saurashtra University, Rajkot, Gujarat - 360005. INDIA. e-mail: samirk- Pn for even n are E-cordial graphs. The same authors in [email protected] [10] proved that the middle graph, total graph and split graph N. B. Vyas is with Department of Mathematics, Atmiya Institute of Technology and Science, Rajkot, Gujarat - 360005, INDIA. e-mail: niravb- of Pn and the composition of Pn with P2 admit E-cordial [email protected] labeling. INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 10
1 u6 Vaidya and Vyas[11] have proved that the mirror graphs of 1 0 even cycle Cn, even path Pn and hypercube Qk are E-cordial v 6 1 u5 v 1 graphs. The same authors in [12] proved that Kn × P2 and 0 0 1 0 0 1 Pn × P2 are E-cordial graphs for even n while Wn × P2 and 0 1 1 0 u1 K1,n × P2 are E-cordial graphs for odd n. 1 0 1 v 1 Definition 1.7: The wheel graph Wn is defined to be the 0 0 0 v 2 join K1 + Cn. The vertex corresponding to K1 is known as v 5 0 1 apex vertex and vertices corresponding to cycle are known 0 u4 1 1 as rim vertices while the edges corresponding to cycle 0 0 are known as rim edges. We continue to recognize apex 1 0 1 1 0 0 u2 of wheel as the apex of respective graphs obtained from wheel. v 4 0 v 3 1 u3 Definition 1.8: The helm Hn is the graph obtained from a 1 wheel W by attaching a pendant edge to each rim vertex. n Figure 1
Definition 1.9: The closed helm CHn is the graph obtained Theorem 2.3: CHn is E - cordial. v v , v ,...,v from a helm Hn by joining each pendant vertex to form a Proof: Let be the apex vertex, 1 2 n be the vertices cycle. of inner cycle and u1, u2,...,un be the vertices of outer cycle of CHn. We note that |V (CHn)| = 2n + 1 and |E(CHn)| = 4n. We define f : E(CHn) → {0, 1} as Definition 1.10: The flower graph F ln is the graph obtained follows: from a helm Hn by joining each pendant vertex to the apex of the helm. For 1 ≤ i ≤ n:
f(vv ) = 1 Definition 1.11: The double triangular snake DTn is i f(v u )=0 obtained from a path Pn with vertices v1, v2,...,vn by i i f(v v )=1 (v = v ) joining vi and vi+1 to a new vertex wi for i = 1, 2,...,n− 1 i i+1 n+1 1 f(u u )=0 (u = u ) and to a new vertex ui for i = 1, 2,...,n− 1. i i+1 n+1 1 In view of above defined labeling pattern f satisfies the Definition 1.12: Let e = uv be an edge of graph G and w vertex and edge conditions for E-cordial labeing as shown in is not a vertex of G. The edge e is subdivided when it is Table II. Hence CHn is E-cordial graph. 0 00 replaced by edges e = uw and e = wv. Illustration 2.4: CH6 and its E-cordial labeling is shown in Figure 2.
Definition 1.13: The gear graph Gn is obtained from the 6 u1 0 u 0 wheel by subdividing each of its rim edges. 0 0 0 1 1 1 II. MAIN RESULTS v 6 v 1 0 0 1 1 1 1 Theorem 2.1: F ln is E - cordial. Proof: Let Hn be a helm with v as the apex vertex, u5 u2 0 v 5 1 2 0 0 1 v v1, v2,...,vn be the vertices of cycle and u1, u2,...,un 0 1 1 0 be the pendant vertices for n > 3. Let F ln be the flower v 1 1 graph obtained from helm Hn then |V (F ln)| = 2n + 1 and 1 1 |E(F ln)| = 4n. We define f : E(F ln) → {0, 1} as follows. 0 0 v 4 v 3 1 ≤ i ≤ n 1 1 For : 1 0 0 0 0 0 f(vvi) = f(viui)=1 u4 u3 f(vu )=0 i Figure 2 f(vivi+1) = 0 (vn+1 = v1) Theorem 2.5: DTn is E - cordial. In view of above defined labeling pattern f satisfies the vertex Proof: Let v1, v2,...,vn be the vertices of path Pn and 0 0 0 and edge conditions for E-cordial labeing as shown in Table u1, u2,...,un and u1, u2,...,un be the newly added vertices I. Hence F ln is E-cordial graph. in order to obtain DTn. We note that |V (DTn)| = 3n − 2 Illustration 2.2: F l6 and its E-cordial labeling is shown in and |E(DTn)| = 5n − 5. We define f : E(DTn) → {0, 1} as in Figure 1. follows: INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 11
Case 1: n ≡ 1(mod 2) For 1 ≤ i ≤ n − 1: f(v u )=1 For 1 ≤ i ≤ n − 1 1 1 f(u1v2)=0 1 i ≡ 1(mod2) 0 i ≡ 1, 2(mod4) f(vv ) = f(v v ) = i 0 otherwise i i+1 1 otherwise 0 i ≡ 1(mod2) f(v u ) = Sub Case 1: n ≡ 1(mod4) i i 1 otherwise f(vvn)=1 0 i ≡ 1(mod2) f(u v ) = For 2 ≤ i ≤ n: i i+1 1 otherwise 0 1 i ≡ 2, 3( mod 4) f(viui)=1 f(v u ) = 0 i i 0 otherwise f(uivi+1)=0 1 i ≡ 2, 3( mod 4) f(u v ) = (consider v = v ) Case 2: n ≡ 0(mod 2) i i+1 0 otherwise n+1 1
Sub Case 2: n ≡ 3(mod4) f(v1v2)=1 f(vvn)=0 f(v u ) = 1 For 2 ≤ i ≤ n − 1 n−1 n−1 f(un−1vn)=1 f(v u )=0 1 i ≡ 0(mod2) n n f(v v ) = f(u v )=0 i i+1 0 otherwise n 1 For 2 ≤ i ≤ n − 2: For 1 ≤ i ≤ n − 1 1 i ≡ 2, 3(mod4) f(v u ) = i i 0 otherwise 0 i ≡ 1(mod2) f(v u ) = 1 i ≡ 2, 3(mod4) i i 1 otherwise f(u v ) = i i+1 0 otherwise 0 i ≡ 1(mod2) f(uivi+1) = 1 otherwise Case 2: n ≡ 0(mod 2) 0 f(viui)=1 0 f(uivi+1)=0 Sub Case 1: n ≡ 0(mod 4):
In view of above defined labeling pattern f satisfies the vertex 0 i ≡ 1(mod2) f(vv ) = and edge conditions for E-cordial labeing as shown in Table i 1 otherwise III. Hence DTn is E-cordial graph. 1 i ≡ 1, 2(mod4) f(viui) = Illustration 2.6: DT5 and its E-cordial labeling is shown 0 otherwise in in Figure 3. 1 i ≡ 1, 2(mod4) f(u v ) = i i+1 0 otherwise u1 0 u2 0 u3 0 u4 0 Sub Case 2: n ≡ 2(mod 4): 1 1 0 0 1 0 0 1 f(vv1) = 1 f(vvn)=0 0 0 1 1 f(v u ) = 1 v1 1 0 1 0 0 n n 2 3 4 5 v v v v f(unv1)=0 For 2 ≤ i ≤ n − 1: 1 0 1 0 1 0 1 0 1 4 1 1 i ≡ 0(mod2) u'1 1 u'2 u'3 1 u' f(vv ) = i 0 otherwise Figure 3 For 1 ≤ i ≤ n − 1: Theorem 2.7: Gn is E - cordial. 1 i ≡ 1, 2(mod4) f(v u ) = Proof: Let Wn be a wheel with apex vertex v and rim vertices i i 0 otherwise v1, v2,...,vn. To obtain the gear graph Gn subdivide each 1 i ≡ 1, 2(mod4) f(u v ) = rim edges of wheel by the vertices u1, u2,...,un. Where i i+1 0 otherwise each ui is added between vi and vi+1 for i = 1, 2,...,n− 1 and un is added between v1 and vn. Then |V (Gn)| = 2n + 1 In view of above defined labeling pattern f satisfies the and |E(Gn| = 3n. We define f : E(Gn) → {0, 1} as follows. vertex and edge conditions for E-cordial labeing as shown in Table IV. Hence Gn is E-cordial graph. Case 1: n ≡ 1(mod 2) Illustration 2.8: G7 and its E-cordial labeling is shown in Figure 4. INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 12
1 v7 u6 1 1 u7 1 0 0 1 v6 1 1 v1 1 0 1 0 1 u1 u5 0 1 0 0 0 1 0 v 1 1 v2 v5 0 1 1 0 0 0 u4 u2 0 1
v4 1 0 1 3 1 1 v u3 Figure 4
CONCLUDING REMARKS Some new E-cordial graphs are investigated. To investigate some characterization(s) or sufficient condition(s) for the graph to be E-cordial is an open area of research.
REFERENCES [1] L. W. Beineke and S. M. Hegde, “Strongly Multiplicative graphs”, Discuss. Math. Graph Theory, vol. 21, pp. 63-75, 2001. [2] I. Cahit, “Cordial Graphs: A weaker version of graceful and harmonious Graphs”, Ars Combinatoria, vol. 23, pp. 201-207, 1987. [3] J. A. Gallian, “A dynamic survey of graph labeling”,The Electronics Journal of Combinatorics,vol.18(#DS6), 2011. Available: http: //www1.combinatorics.org/Surveys/ds6.pdf [4] S. W. Golomb, “How to number a graph”, in Graph theory and Computing, R. C. Read, ed., Academic Press, New York, pp. 23-37, 1972. [5] J. Gross and J. Yellen, “Graph Theory and its Applications”, CRC Press, 1999 [6] S. Lo, “On edge graceful labelings of graphs”, Congr. Numer., vol. 50, pp. 231-241, 1985. [7] G. Ringel, “Problem 25, Theory of Graphs and its Applications”, Proc. Symposium Smolenice, 1963, Prague, 162, 1964. [8] A. Rosa, “On certain valuations of the vertices of a graph”, Theory of graphs ( Internat. Symposium, Rome, July 1966 ), Gordon and Breach, N. Y. and Dunod Paris, 349-355, 1967. [9] S. K. Vaidya and Lekha Bijukumar, “Some New Families of E-cordial Graphs”, Journal of Mathematics Research, vol. 3 , pp. 105-111, 2011. doi:10.5539/jmr.v3n4p105 [10] S. K. Vaidya and Lekha Bijukumar, “Some New Results on E-cordial Labeling”, Int. Journal of Information Science and Computer Mathemat- ics, vol. 3 , pp. 21-29, 2011. Available: http://www.pphmj.com/abstract/ 6004.htm [11] S. K. Vaidya and N. B. Vyas, “E-cordial Labeling of Some Mirror graphs”, Int. Journal of Contemporary Advanced Mathematics , vol. 2, pp. 22-27, 2011. Available: http://www.cscjournals.org/csc/manuscript/ Journals/IJCM/volume2/Issue1/IJCM-19.pdf [12] S. K. Vaidya and N. B. Vyas, “E-cordial Labeling for Cartesian Product of Some Graphs”, Studies in Mathematical Sciences, vol. 2, pp. 11-15, 2011. doi:10.3968/j.sms.1923845220110302.175 [13] R. Yilmaz and I. Cahit, “E-cordial graphs”, Ars Combin., vol. 46, pp. 251-266, 1997. INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 13
TABLE I
Vertex Condition Edge Condition
n ≡ 0(mod 2) vf (0) − 1 = vf (1) = n ef (0) = ef (1) = 2n
n ≡ 1(mod 2) vf (0) = vf (1) − 1 = n ef (0) = ef (1) = 2n
TABLE II
Vertex Condition Edge Condition
n ≡ 0(mod 2) vf (0) − 1 = vf (1) = n ef (0) = ef (1) = 2n
n ≡ 1(mod 2) vf (0) = vf (1) − 1 = n ef (0) = ef (1) = 2n
TABLE III
Vertex Condition Edge Condition
3n−2 5n−5 n ≡ 0(mod 2) vf (0) = vf (1) = 2 ef (0) − 1 = ef (1) = b 2 c 3n−2 5n−5 n ≡ 1(mod 4) vf (0) − 1 = vf (1) = b 2 c ef (0) = ef (1) = 2 3n−2 5n−5 n ≡ 3(mod 4) vf (0) = vf (1) − 1 = b 2 c ef (0) = ef (1) = 2
TABLE IV
Vertex Condition Edge Condition
3n+1 n ≡ 1(mod 2) vf (0) = vf (1) − 1 = n ef (0) + 1 = ef (1) = 2 3n n ≡ 0(mod 2) vf (0) − 1 = vf (1) = n ef (0) = ef (1) = 2 Annals of Pure and Applied Mathematics Annals of Vol. 3, No. 2, 2013, 119-128 ISSN: 2279-087X (P), 2279-0888(online) Published on 28 July 2013 www.researchmathsci.org
Antimagic Labeling of Some Path and Cycle Related Graphs
S. K. Vaidya 1 and N. B. Vyas 2 1Department of Mathematics, Saurashtra University, Rajkot – 360005, Gujarat, India. E-mail: [email protected] 2Department of Mathematics, Atmiya Institute of Technology and Science, Rajkot – 360005, Gujarat, India. E-mail: [email protected]
Received 29 June 2013; accepted 23 July 2013
Abstract. An edge labeling of a graph is a bijection from E(G) to the set {1,2,… , |E(G)|}. If for any two distinct vertices u and v, the sum of labels on the edges incident to u is different from the sum of labels on the edges incident to v then an edge labeling is called antimagic labeling. We investigate antimagic labeling for some path and cycle related graphs.
Keywords: Antimagic labeling, Antimagic graph, Middle graph, Total graph.
AMS Mathematics Subject Classification (2010): 05C78
1. Introduction We begin with a finite, connected and undirected graph G=(V(G),E(G)) without loops and multiple edges. Throughout this paper |V(G)| and |E(G)| denote the number of vertices and number of edges respectively. For any graph theoretic notation and terminology we rely upon Balakrishnan and Ranganathan [1]. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (edges) then the labeling is called a vertex labeling (an edge labeling). According to Beineke and Hegde[2] labeling of discrete structure is a frontier between graph theory and theory of numbers. For an extensive survey of graph labeling as well as bibliographic references we refer to Gallian[3]. The concept of magic labeling was introduced during 1963 by Sedlacek[5]. A graph is said to be magic if it has a real-valued edge labeling such that; (i) distinct edges have distinct non-negative labels; (ii) the sum of the labels of the edges incident to a particular vertex is same for all the vertices.
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S. K. Vaidya and N. B. Vyas An antimagic labeling of a graph G is a bijection from E(G) to the set {1, 2, … , |E(G)|} such that for any two distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. Hartsfield and Ringel[4] have introduced the concept of an antimagic graph in 1990. They proved that paths Pn (n ≤ 3), cycles, wheels, and complete graphs Kn (n ≤ 3) admit antimagic labeling. They have also conjectured that; (i) all trees except K2 are antimagic. (ii) all connected graphs except K2 are antimagic. These two conjectures are still not settled. The graphs obtained by switching of vertex in path Pn, cycle Cn, wheel Wn, helm Hn and fan fn are proved to be antimagic by Vaidya and Vyas[6]. The middle graph M(G) of a graph G is the graph whose vertex set is V(G) ∪ E(G) and in which two vertices are adjacent if and only if either they are adjacent edges of G or one is a vertex of G and the other is an edge incident on it. The total graph T(G) of a graph G is the graph whose vertex set is V(G) ∪ E(G) and two vertices are adjacent whenever they are either adjacent or incident in G. For a graph G the splitting graph S'(G) is obtained by adding a new vertex v' corresponding to each vertex v of G such that N(v)=N(v') where N(v) and N(v') are the neighborhood sets of v and v' respectively. The shadow graph D2(G) of a connected graph G is constructed by taking two copies of G, say G' and G''. Join each vertex u' in G' to the neighbours of the corresponding vertex u'' in G''.
2. Main Results
Theorem 2.1. Middle graph of path Pn is antimagic. Proof. Let v1, v2, ... , vn be the vertices and e1, e2, ... , en be the edges of path Pn and G = M(Pn) be the middle graph of path Pn. According to the definition of middle graph V(M(Pn)) = V(Pn) ∪ E(Pn) and E(M(Pn)) = {viei ; 1≤ i ≤ n – 1, viei-1 ; 2≤ i ≤ n, eiei+1 ; 1≤ i ≤ n – 2}. Here |V(G)| = 2n – 1 and |E(G)| = 3n – 4. To define f : E(G) → { 1, 2, … , 3n – 4 }, we consider following two cases.
Case 1: n = 3, 5 and n ≡ 0 (mod 2) For 1 ≤ i ≤ n – 1: f (viei) = 2i – 1 ; f (eivi+1) = 2i; For 1 ≤ i ≤ n – 2: f (eiei+1) = 2(n – 1 ) + i ;
Case 2: n ≡ 1 (mod 2) where n > 5 For 1 ≤ i ≤ n – 1: f (viei) = 2i – 1 ; f (eivi+1) = 2i; For 1 ≤ i ≤ n – 4: f (eiei+1) = 2(n – 1) + i; f (en-3en-2) = 3n– 4; f (en-2en-1) = 3n– 5;
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Above defined edge labeling function will generate all the distinct vertex labels satisfying the condition for antimagic labeling. Hence M(Pn) is antimagic.
Illustration 2.2. Middle graph of path P5 and its antimagic labeling is shown in Figure 1.
Figure 1
Theorem 2.3. Middle graph of cycle Cn is antimagic. Proof. Let v1, v2, ... , vn be the vertices and e1, e2, ... , en be the edges of cycle Cn and G = M(Cn) be the middle graph of cycle Cn. According to the definition of middle graph V(M(Cn)) = V(Cn) ∪ E(Cn) and E(M(Cn)) = {vi ei ; 1 ≤ i ≤ n, vi ei-1 ; 2 ≤ i ≤ n, ei ei+1 ; 1≤ i ≤ n – 1,en e1}. Here |V(G)| = 2n and |E(G)| = 3n. To define f : E(G) →{1, 2, … 3n}, we consider following two cases.
Case 1: n ≡ 1 (mod 2) For 1 ≤ i ≤ n : f (viei) = 2i ; For 1 ≤ i ≤ n – 1 : f (eivi+1) = 2i + 1 ; f (env1) = 1; f (eiei+1) = 2n + 1 + i ; f (ene1) = 2n + 1;
Case 2: n ≡ 0 ( mod 2) f (v1e1) = 2 ; For 2 ≤ i ≤ n: f (viei) = 2i ; For 1 ≤ i ≤ n – 1: f (eivi+1) = 2i + 1 ; f (env1) = 1 ; For 1 ≤ i ≤ n – 1: f (eiei+1) = 2n + 1 + i ; f (ene1) = 2n + 1 ;
Above defined edge labeling function will generate all the distinct vertex labels satisfying the condition for antimagic labeling. Hence M(Cn) is antimagic.
Illustration 2.4. Middle graph of cycle C5 and its antimagic labeling is shown in Figure 2.
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Figure 2
Therom 2.5. Total graph of path Pn is antimagic. Proof. Let v1, v2, ... , vn be the vertices and e1, e2, ... , en-1 be the edges of path Pn and G = T(Pn) be the total graph of path Pn with V(T(Pn)) = V(Pn) ∪ E(Pn) and E(T(Pn)) = {vi vi+1 ; 1≤ i ≤ n – 1, vi ei ; 1 ≤ i ≤ n – 1, ei ei+1 ; 1 ≤ i ≤ n – 2, vi ei-1; 2 ≤ i ≤ n}. Here |V(G)| = 2n – 1 and |E(G)| = 4n – 5 . Define f :E(G) →{1, 2, … 4n – 5} as follows.
For 1 ≤ i ≤ n – 2: f (vivi+1) = 4i ; f (vn-1vn) = 4n – 5 ; f (eiei+1) = 4i – 3 ; For 1 ≤ i ≤ n – 2: f (viei) = 4i – 1; For 2 ≤ i ≤ n – 1: f (viei-1) = 4i – 2 ; f (vnen-1) = 4n – 7 ;
Above defined edge labeling function will generate all the distinct vertex labels satisfying the condition for antimagic labeling. Hence T(Pn) is antimagic.
Illustration 2.6. Total graph of path T(P6) and its antimagic labeling is shown in Figure 3.
Figure 3
Theorem 2.7. Total graph of cycle Cn is antimagic. Proof. Let v1, v2, ... , vn be the vertices and e1, e2, ... , en be the edges of cycle Cn and G=T(Cn) be the total graph of cycle Cn with V(T(Cn))=V(Cn) ∪ E(Cn) and E(T(Cn)) = {vivi+1 ;1≤ i ≤ n – 1, vn v1,vi ei ;1 ≤ i ≤ n, ei ei+1 ;1 ≤ i ≤ n – 1, en e1, vi ei+1;2≤ i ≤ n, v1en}. Here |V(G)| = 2n and |E(G)| = 4n. To define f:E(G) →{1, 2, … 4n} we consider following two cases.
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Case 1: n ≡ 1 (mod 2) For 1 ≤ i ≤ n – 1: f (vivi+1) = 3n+i ; f (vnv1) = 4n ; For 2 ≤ i ≤ n – 1: f (eiei+1) = i + 1 ; f (ene1) = 1 ; For 1 ≤ i ≤ n : f (viei) = n +2i ; For 2 ≤ i ≤ n : f (viei-1) = n –1+2i ; f (v1en) = n +1 ;
Case 2: n ≡ 0 (mod 2) Sub case 1: n ≡ 0 (mod 4) For 1 ≤ i ≤ n – 1: f (vivi+1) = 3n+i ; f (vnv1) = 4n ; For 2 ≤ i ≤ n – 1: f (eiei+1) = i+1 ; f (ene1) = 2 ; f (e1e2) = 1 ; For 1 ≤ i ≤ n : f (viei) = n+2i ; For 2 ≤ i ≤ n : f (viei-1) = n – 1 + 2i ; f (v1en) = n + 1 ;
Sub case 2: n ≡ 2 (mod 4) f (v1v2) = 3n + 2 ; f (v2v3) = 3n + 1 ; For 3 ≤ i ≤ n – 1: f (vivi+1) = 3n + i ; For 2 ≤ i ≤ n – 1: f (eiei+1) = i +1 ; f (ene1) = 2 ; f (e1e2) = 1 ; For 1 ≤ i ≤ n : f (viei) = n +2i ; For 2 ≤ i ≤ n : f (viei-1) = n – 1 +2i ; f (v1en) = n + 1;
Above defined edge labeling function will generate all the distinct vertex labels satisfying the condition for antimagic labeling. Hence T(Cn) is antimagic.
Illustration 2.8. Total graph of cycle C6 and its antimagic labeling is shown in Figure 4.
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Figure 4 Theorem 2.9. Splitting graph of path Pn is antimagic. Proof. Let v1, v2, ... , vn be the vertices and e1, e2, ... , en-1 be the edges of path Pn. Let
vv12′′,,,… vn ′ be the newly added vertices to form the splitting graph of path Pn. Let G=
S(Pn) be the splitting graph of path Pn . V(S'(Pn)) = {,vvii′ /1≤ i≤ n } and E(S'(Pn)) ′′ ={vvii+−+111 ;1≤≤ in − 1, vv ii ;2 ≤≤ invv , ii ;1 ≤≤ in − 1} . Here |V(G)| = 2n and |E(G)| = 3n – 3. Define f :E(G) → {1, 2, … , 3n – 3} as follows.
For 1 ≤ i ≤ n – 1:
f (vvii+1 ) = 3 i;
f ()32vvii′ +1 =− i ;
f ()31vvii′+1 =− i ;
Above defined edge labeling function will generate all the distinct vertex labels satisfying the condition for antimagic labeling. Hence S'(Pn) is antimagic.
Illustration 2.10. Splitting graph of path P6 and its antimagic labeling is shown in Figure 5.
Figure 5
Theorem 2.11. Splitting graph of cycle Cn is antimagic.
Proof. Let vv12,,,… vn be the vertices and ee12,,,… en be the edges of cycle Cn. Let
vv12′′,,,… vn ′ be the newly added vertices to form the splitting graph of cycle Cn. Let
G=S'(Cn) be the splitting graph of cycle Cn. V(S'(Cn)) = {vvii ,′ /1≤ i≤ n } and E(S'(Cn))
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={vvii′+−+111111 ;1≤≤ i n − 1, v n′′′ v , vv n , vv ii ;2 ≤≤ i n , vv ii ;1 ≤≤ i n − 1, v n v } . Here |V(G)| = 2n and |E(G)| = 3n. To define f :E(G) →{1, 2, … , 3n} we consider following two cases.
Case 1: nmod≡ 1( 2) For 1 ≤ i ≤ n – 1 :
f ()21vvii+1 =++ n i; f ()2vvn 1 = n; ⎧2ii+≡ 1, 1(mod 2); fvv()ii′ +1 = ⎨ f ()21vvn′ 1 = n+ ; ⎩2,i otherwise . For 2 ≤ i ≤ n: ⎧21,1(mod2);ii−≡ fvv()ii′ −1 = ⎨ fvv()11′ n = ; ⎩22,i− otherwise .
Case 2: n ≡ 0 (mod 2) For 1 ≤ i ≤ n – 1 :
f ()21vvii+1 =++ n i; f ()3vvn 1 = n;
Sub Case 1: n ≡ 0 (mod 4), n ≠ 4
fvv()421′ = ; fvv()223′ = ; For 1 ≤ i ≤ n, (i ≠ 2): ⎧2ii+≡ 1, 1(mod 2); fvv()ii′ +1 = ⎨ fvv()11′ n = ; ⎩2,i otherwise . For 3 ≤ i ≤ n : ⎧21,1(mod2);ii−≡ fvv()ii′ −1 = ⎨ ⎩22,i− otherwise .
Sub Case 2: n = 4 and n ≡ 1 (mod 4)
fvv()31′ n = ; fvv()112′ = ; For 2 ≤ i ≤ n: ⎧2ii+≡ 1, 1(mod 2); ⎧21,1(mod2);ii−≡ fvv()ii′ +1 = ⎨ fvv()ii′ −1 = ⎨ ⎩2,i otherwise . ⎩22,i− otherwise .
Above defined edge labeling function will generate all the distinct vertex labels satisfying
the condition for antimagic labeling. Hence SC′()n is antimagic.
Illustration 2.12. Splitting graph of cycle C4 and its antimagic labeling is shown in Figure 6.
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Figure 6
Theorem 2.13. Shadow graph of path Pn is antimagic.
Proof. Let Pn′ , Pn′′ be two copies of path Pn. We denote the vertices of first copy of Pn by
vv12′′, ,...., vn ′ and second copy by vv12′′, ′′ ,...., vn ′′ . Let G be DP2 ()n with |V(G)| = 2n and |E(G)| = 4n – 4. Define f : E(G) →{1, 2, … , 4n – 4 } as follows.
For 1 ≤ i ≤ n – 1 :
f ()4vvii′′+1 = i; f ()43vvii′′ ′′+1 = i− ; f ()41vvii′ ′′+1 = i− ; For 2 ≤ i ≤ n:
f ()42vvii′′′−1 =− i ;
Above defined edge labeling function will generate all distinct vertex labels satisfying the
condition for antimagic labeling. Hence DP2 ()n is antimagic.
Illustration 2.14. Shadow graph of path P6 and its antimagic labeling is shown in Figure 7.
Figure 7
Theorem 2.15. Shadow graph of cycle Cn is antimagic.
Proof. Let Cn′ , Cn′′ be two copies of cycle Cn. We denote the vertices of first copy
of Cn by vv12′′,,,… vn ′and second copy by vv12′′, ′′ ,...., vn ′′ . Let G be DC2 ()n with |V(G)| = 2n and |E(G)| = 4n. To define f : E(G) → {1, 2, … , 4n} we consider following three cases.
Case 1: n ≡ 1(mod 2)
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For 1 ≤ i ≤ n – 1 :
f ()vvii′′ ′′+1 = i; f ()vvn′′′′1 = n;
f ()3vvii′′+1 =+ n i; f ()4vvn′ 1′ = n;
f ()vvii′′ ′+1 =+ n 21 i −; f ()31vvn′′′1 = n− ;
f ()vvii′′′−1 =+ n 2 i; f ()3vvn′ 1′′ = n;
Case 2: n ≡ 0(mod 2), n ≠ 6
fvv()1n′′1 ′′ = ; f ()4vvn′ 1′ = n;
f ()vvn′′′1 =+ n 1; f ()3vv1′ n′′ = n; For 1 ≤ i ≤ n – 1:
f ()3vvii′′ ′′+1 =+ n i; f ()1vvii′ ′+1 = i+ ;
f ()vvii′′′+1 =++ n 12 i; For 2 ≤ i ≤ n:
f ()vvii′′′−1 =+ n 2 i;
Case 3: For n = 6, antimagic labeling of D2(C6) is shown in below Figure 8.
Figure 8
Above defined edge labeling function will generate all the distinct vertex labels satisfying
the condition for antimagic labeling. Hence DC2 ()n is antimagic.
Illustration 2.16. Shadow graph of cycle C5 and its antimagic labeling is shown in Figure 9.
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Figure 9 3. Concluding Remarks We have investigated antimagic labeling for shadow graph, middle graph and total graph of Pn and Cn. More exploration is possible for other graph families and in the context of different graph labeling problems.
REFERENCES
1. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Springer, 2012. 2. L. W. Beineke and S. M. Hegde, Strongly Multiplicative graphs, Discuss. Math. Graph Theory, 21, (2001), 63-75. 3. J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronics Journal of Combinatorics, 19 (#DS6), 2012. Available: http://www.combinatorics.org/Surveys/ds6.pdf 4. N. Hartsfield and G. Ringel, Pearls in Graph Theory: A Comprehensive Introduction, Academic Press, Boston, 1990, 108-110. 5. J. Sedlacek, Problem 27. in Theory of Graphs and its Applications, Proc. Symposium Smolenice, June, (1963), 163 - 167. 6. S. K. Vaidya and N. B. Vyas, Antimagic Labeling in the Context of Switching of a Vertex, Annals of Pure and Applied Mathematics, 2(1), (2012), 33-39. Available: www.researchmathsci.org/apamart/apam-v2n1-4.pdf
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ISSN 2320 – 6543 Volume 1 Issue 4 August 2013 pp.122-130
EVEN MEAN LABELING FOR PATH AND BISTAR RELATED GRAPHS
S. K. Vaidya a,* and N. B. Vyas b a Department of Mathematics, Saurashtra University, Rajkot - 360005, Gujarat, India b Atmiya Institute of Technology and Science, Rajkot - 360005, Gujarat, India
Corresponding author: S. K. Vaidya Tel.: +91-9825292539; e-mail: [email protected] [Received 28/06/2013 | Received in revised form 31/07/2013 | Accepted 31/07/2013]
ABSTRACT An even mean labeling is a variant of mean labeling. Here we investigate even mean labeling for path and bistar related graphs.
Keywords: Shadow graph; Splitting graph; Middle graph; Even Mean labeling. 2010 Mathematics Subject Classification: 05C78.
1. INTRODUCTION
Throughout this work, by a graph we mean finite, connected, undirected, simple graph G = (V(G),E(G)) of order |V(G)| and size |E(G)|.
A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (edges) then the labeling is called a vertex labeling (an edge labeling).
According to Beineke and Hegde[1] labeling of discrete structure is a frontier between graph theory and theory of numbers. A latest survey on various graph labeling problems can be found in Gallian[2].
The function f is called mean labeling of graph G if f :V(G) →{0, 1, 2, … , |E(G)|} is injective and the induced function f *: E(G) →{1, 2, … , |E(G)|} defined as f()() u f v ,if f ( u ) f ( v ) is even; f* () e uv 2 is bijective. A graph which admits mean labeling f( u ) f ( v ) 1 ,if f ( u ) f ( v ) is odd. 2 is called a mean graph.
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Even Mean Labeling for Path and Bistar Related graphs
The mean labeling was introduced by Somasundaram and Ponraj [3] and they proved that the graphs Pn, Cn, Pn × Pm, Pm × Cn are mean graphs. Vaidya and Lekha [4] proved that the graphs Pm 2 [P2], Pn , M(Pn) and some cycle related graphs admit mean labeling.
According to Pricilla [5], function f is called even mean labeling of graph G if f : V(G) →{ f()() u f v 2,4, …, 2|E(G)|} is injective and each edge uv assigned the label such that the resulting 2 edge labels are distinct.
For a connected graph G, let G be the copy of G then shadow graph D2(G) is obtained by joining each vertex u in G to the neighbours of the corresponding vertex u in G . The middle graph M(G) of a graph G is the graph whose vertex set is VGEG()() and in which two vertices are adjacent if and only if either they are adjacent edges of G or one is a vertex of G and the other is an edge incident on it. The total graph T(G) of a graph G is the graph whose vertex set is VGEG()() and two vertices are adjacent whenever they are either adjacent or incident in G. For a graph G the splitting graph SG()is obtained by adding new vertex v corresponding to each vertex v of G such that N()() v N v where N(v) and Nv() are the neighbourhood sets of v and v respectively.
Duplication of a vertex vk by a new edge e vkk v in a graph G produces a new graph such that
N()() vk N v k v k . A vertex switching Gv of a graph G is the graph obtained by taking a vertex v of G, removing all the edges to v and adding edges joining v to every other vertex which are not adjacent to v in G. Square of a graph G denoted by G2 has the same vertex set as of G and two vertices are adjacent in G2 if they are at a distance at most 2 apart in G. Cube of a graph G denoted by G3 has the same vertex set as of G and two vertices are adjacent in G3 if they are at a distance at most 3 apart in
G. The double fan DFn is obtained by Pn+2K1. For any undefined term in graph theory we rely upon West [6].
2. RESULTS
Theorem 2.1. The graph DP2 ()n is an even mean graph.
Proof. Let v12,,, v vn be the vertices of path Pn and v12,,, v vn be the newly added vertices corresponding to the vertices v12,,, v vn in order to obtain DP2 ()n . Denoting GDP2 ()n then |V ( G )| 2 n and |E ( G )| 4( n 1) . We define f: V ( G ) {2,4, ,2| E ( G )|} as follows. For 1in : 8ii 6, 1(mod 2); fv() i 8i 12, otherwise. 8ii , 1(mod 2); fv() i 8i 10, otherwise.
The above defined function f provides an even mean labelling for DP2 ()n . That is, is an even mean graph. □
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Illustration 2.2. Shadow graph of path P5 and its even mean labeling is shown in Fig.1.
Fig.1
Theorem 2.3. Middle graph of path Pn is an even mean graph.
Proof. Let v12, v ,..., vn be the vertices and e1, e 2 ,..., en 1 be the edges of path Pn and GMP()n be the middle graph of path Pn. According to the definition of middle graph VMPVPEP( (n )) ( n ) ( n ) and
EMP( (n )) = {vei i ;1 in 1, ve i i11 ;2 inee , i i ;1 in 2}. Here |V ( G )| 2 n 1 and |E ( G )| 3 n 4. We define f: V ( G ) {2,4, ,2| E ( G )|} as follows. For 1in : f( vi ) 4 i 2 ; For 1in 1: f( ei ) 4 i ;
The above defined function f provides an even mean labeling for MP()n . Hence, MP()n is an even mean graph. □
Illustration 2.4. MP()5 and it’s even mean labeling is shown in Fig.2.
Fig.2
Theorem 2.5. Total graph of path Pn is an even mean graph.
Proof. Let v12,,, v vn be the vertices and e1,,, e 2 en 1 be the edges of path Pn and GTP()n be the total graph of path Pn with VTP( (n )) = VPEP()()nn and ETP( (n )) = {vi v i1 ;1 i n 1, v i e i ;1 i n 1, ei e i11;1 i n 2, v i e i ;2 i n } . Here |V ( G )| 2 n 1 and |E ( G )| 4 n 5 . We define f: V ( G ) {2,4, ,2| E ( G )|} as follows. For 1in : f( vi ) 4 i 2 ; For 1in 1:
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f( ei ) 4 i ;
The above defined function f provides an even mean labeling for TP()n . Hence, TP()n is an even mean graph. □
Illustration 2.6. TP()6 and its even mean labeling is shown in Fig.3.
Fig.3
Theorem 2.7. Splitting graph of path Pn is an even mean graph.
Proof. Let v12,,, v vn be the vertices and e1,,, e 2 en 1 be the edges of path Pn. Let v12,,, v vn be the newly added vertices to form the splitting graph of path Pn. Let GSP()n be the splitting graph of path Pn. VSP( (n )) = {vii } { v },1 i n and ESP( (n )) = {vi v i1 ;1 i n 1, v i v i 1 ;2 i n , v i v i 1 ;1in 1}. Here |V ( G )| 2 n and |E ( G ))| 3 n 3. We define f: V ( G ) {2,4, ,2| E ( G )|} as follows. For 1in : 4ii , 1(mod 2); fv() i 4i 2, otherwise. 4ii 2, 1(mod 2); fv() i 4i , otherwise.
The above defined function f provides an even mean labeling for SP()n . Hence, is an even mean graph. □
Illustration 2.8. SP()6 and its even mean labeling is shown in Fig.4.
Fig.4
Theorem 2.9. Duplicating each vertex by an edge in path Pn is an even mean graph.
Proof. Let v12,,, v vn be the vertices of path Pn. Let G be the graph obtained by duplicating each vertex vi of Pn by an edge vviiat a time , where 1 ≤ i ≤ n. Note that |V ( G ) | 3 n and |E ( G )| 4 n 1. We define f: V ( G ) {2,4, ,2| E ( G )|} as follows.
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For 1in : f( vi ) 8 i 6; f( vi ) 8 i 4; f( vi ) 8 i 2; The above defined function f provides an even mean labeling for graph G. Hence, duplicating each vertex by edge in path Pn is an even mean graph. □
Illustration 2.10. Duplicating each vertex by edge in path P7 and its even mean labeling is shown in Fig.5.
Fig.5
Theorem 2.11. Switching of a pendant vertex in path Pn is an even mean graph.
Proof. Let v12, v ,..., vn be the vertices of Pn and Gv denotes the graph obtained by switching of a pendant vertex v of GPn . Without loss of generality let the switched vertex be v1. We note that |V ( G ) | n and |E ( G ) | 2 n 4 . v1 v1 We define f: V ( G ) {2,4,...,4 n 8} as follows: v1 fv(1 ) 2 ; For 2in : f( vi ) 4 n 4 2 i ; The above defined function f provides an even mean labeling for G . Hence, the graph v1 obtained by switching of a pendant vertex in a path Pn is an even mean graph. □
Illustration 2.12. Switching of a pendant vertex in path P5 and its even mean labeling is shown in Fig.6.
Fig.6
2 Theorem 2.13. Pn is an even mean graph. 2 Proof. Let v12,,, v vn be the vertices of path Pn. Let GPn then note that |V ( G )| n and |E ( G )| 2 n 3. We define f: V ( G ) {2,4, ,4 n 6} as follows.
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For 1in : f( vi ) 2 i ; 2 2 The above defined function f provides an even mean labeling for Pn . Hence, Pn is an even mean graph. □
2 Illustration 2.14. P6 and its even mean labeling is shown in Fig.7.
Fig.7
3 Theorem 2.15. Pn is an even mean graph. 3 Proof. Let v12,,, v vn be the vertices of path Pn. Let GPn then note that |V ( G )| n and |E ( G )| 3 n 6. We define f: V ( G ) {2,4, ,6 n 12} as follows. For 1in : i 12 10,i 1(mod4 ); 4 i 12 8,i 2(mod4 ); 4 fv() i i 12 6,i 3(mod4 ); 4 i 12 2,i 0(mod4 ). 4
3 3 The above defined function f provides an even mean labeling for Pn . Hence, Pn is an even mean graph. □
3 Illustration 2.16. P5 and it’s even mean labeling is shown in Fig.8.
Fig.8
2 Theorem 2.17. Bnn, is a mean graph.
Proof. Consider Bnn, with vertex set {u , v , uii , v ,1 i n } where uvii, are pendant vertices. Let G be
2 the graph Bnn, . Then |V ( G )| 2 n 2 and |E ( G )| 4 n 1. We define f: V ( G ) {2,4, ,8 n 2} as follows. f( u ) 8 n 2 ;
ISSN 2320 – 6543 ijogt, 1(4) (2013), 122-130 Page 127 S. K. Vaidya and N. B. Vyas /International Journal of Graph Theory fv( ) 2 ; For 1in : f( vi ) 2 2 i ; f( ui ) 8 n 2 2 i ; 2 2 The above defined function f provides an even mean labeling for Bnn, . Hence, Bnn, is an even mean graph. □
2 Illustration 2.18. B5,5 and its even mean labeling is shown in Fig.9.
Fig.9
Theorem 2.19. SB()nn, is an even mean graph.
Proof. Consider Bnn, with the vertex set {u , v , uii , v ,1 i n } where uvii, are pendant vertices. In order to obtain SB()nn, , add u,,, v uii v vertices corresponding to u, v, ui, vi where, 1 ≤ i ≤ n. If GSB()nn, then |V ( G )| 4( n 1) and |E ( G )| 6 n 3. We define f: V ( G ) {2,4, ,12 n 6} as follows. f( u ) 6 n 2 ; f( v ) 12 n 6; fu( ) 2 ; f( v ) 6 n 4 ; For 1in : f( ui ) 2 i 2 ; f( ui ) 2 n 2 2 i ; f( vi ) 6 n 4 2 i ; f( vi ) 8 n 4 2 i ;
The above defined function f provides an even mean labeling for SB()nn, . Hence, SB()nn, is an even mean graph. □
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Illustration 2.20. SB()4,4 and it’s even mean labeling is shown in Fig.10.
Fig.10
Theorem 2.21. DFn is an even mean graph.
Proof. Let v12,,, v vn be the vertices of Pn for n even. Vertices u and v are added to obtain
DFnn P2 K1 . We note that |V ( DFn ) | n 2 and |E ( DFn ) | 3 n 1.
We define f: V ( DFn ) {2,4, ,6 n 2} as follows. fu( ) 2; f( v ) 6 n 2 ; For 1in : f( vi ) 2 n 2 i 2 ;
The above defined function f provides an even mean labeling for DFn . Hence, DFn is an even mean graph. □
Illustration 2.22. DF6 and it’s even mean labeling is shown in Fig.11.
Fig.11
CONCLUSIONS
It is always interesting to find out graph or graph families which admit a particular labeling. Here we investigate some new graph families which admit even mean labeling. To investigate similar results for other graph families is an open area of research.
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REFERENCES
[1] L. W. Beineke and S. M. Hegde, Strongly Multiplicative graphs, Discuss. Math. Graph Theory, vol. 21, 63-75, 2001. [2] J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronics Journal of Combinatorics, vol. 19 (#DS6), 2012. URL: http://www.combinatorics.org/Surveys/ds6.pdf [3] S. Somasundaram and R. Ponraj, Some results on mean graphs, Pure and Applied Mathematical Sciences , vol. 58, 29-35, 2003. [4] S. K. Vaidya and Lekha Bijukumar, Some New families of mean graph, Journal of Mathematics Research, vol. 2(3)(2010), 169-176. URL: http://www.ccsenet.org/journal/index.php/jmr/ [5] B. Nirmala Gnanam Pricilla, A Study On New Classes Of Graphs In Variations Of Graceful Graph, Ph.D. Thesis, Bharath University, Chennai, 2008. URL: http://shodhganga.inflibnet.ac.in/handle/10603/33 [6] D. B. West, Introduction to Graph Theory, Prentice-Hall, 2003.
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