Information in Sciences and Computing Volume 2016, Article ID ISC650716, 09 pages Available online at http://www.infoscicomp.com/
Research Article
Even Vertex Equitable Even Labeling of Path and Bistar Related Graphs A. Lourdusamya and F. Patricka a Department
of Mathematics, St.Xavier’s College, Palayamkottai - 627002, Tamil-
nadu, India Corresponding author: A. Lourdusamy, E-mail:
[email protected] Received 26 July 2016; Accepted 04 October 2016 Abstract:Let G be a graph with p vertices and q edges and A = {0, 2, 4, · · · , q + 1} if q is odd or A = {0, 2, 4, · · · , q} if q is even. A graph G is said to be an even vertex equitable even labeling if there exists a vertex labeling f : V (G) → A that induces an edge labeling f ∗ defined by f ∗ (uv) = f (u) + f (v) for all edges uv such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the induced edge labels are 2, 4, · · · , 2q, where vf (a) be the number of vertices v with f (v) = a for a ∈ A. A graph that admits even vertex equitable even labeling is called an even vertex equitable even graph. In 2 this paper, we prove that Pn ⊙ mK1 , Pn (Qm ), S ∗ (Pn ⊙ K1 ), S ∗ (Ln ), S ∗ (Bn,n ), Bn,n ′ and S (Bn,n ) admit an even vertex equitable even labeling. Keywords:Vertex equitable labeling; Even vertex equitable even labeling.
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Introduction
All graphs considered here are simple, finite, connected and undirected. We follow the basic notations and terminologies of graph theory as in [2]. A labeling of a graph is a map that carries the graph elements to the set of numbers, usually to the set of non-negative or positive integers. If the domain is the set of vertices the labeling is called vertex labeling. If the domain is the set of edges, then we speak about edge labeling. If the labels are assigned to both vertices and edges then the labeling is called total labeling. For a dynamic survey of various graph labeling, we refer to Gallian [1]. The vertex set and the edge set of a graph are denoted by V (G) and E(G) respectively. The concept of even vertex equitable even labeling was due to Lourdusamy et al. in [3]. In [3, 4, 5], Lourdusamy et al. proved that path, comb, complete bipartite, cycle, K2 + mK1 , bistar, ladder, S(Ln ), S(Bn,n ), ′ Ln ⊙ K1 , Pn2 , S(Pn ⊙ K1 ), S (Pn ), T (Pn ), graph obtained by duplication of each
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vertex byDan edge in P En , Qn , S(Qn ), D(Qn ), A(Tn ), DA(Tn ), Ln ⊙ mK1 , Cn ⊙ K1 , (1) (2) Pn @Pm , K1,n ∆K1,n , T Ln , Pn ⊙2K1 , T Ln ⊙K1 , Tn ⊙K1 , Qn ⊙K1 , DA(Qn )⊙K1 , DA(Tn ) ⊙ K1 , S(DA(Tn )), S(DA(Qn )), hBn,n : wi and KY (m, n) admit an even vertex equitable even labeling. In [6, 7], Lourdusamy et al. proved that Tp -tree, T oˆPn , T oˆ2Pn , T oˆCnh(n ≡ 0, i3(mod 4)), T o˜Cn (n ≡ 0, 3(mod 4)), T oˆK1,n , T ⊙ Kn , (2)
Cm ⊖ Pn , Cn (Qm ), Pn ; Cm , Cm ∗e Cn and the graph obtained by duplicating an arbitrary vertex and edge of a cycle Cn admit an even vertex equitable even labeling. 2 and Here, we prove that Pn ⊙ mK1 , Pn (Qm ), S ∗ (Pn ⊙ K1 ), S ∗ (Ln ), S ∗ (Bn,n ), Bn,n ′ S (Bn,n ) admit an even vertex equitable even labeling. Definition 1.1 Let G be a graph with p vertices and q edges and A = {0, 2, 4, · · · , q+ 1} if q is odd or A = {0, 2, 4, · · · , q} if q is even. A graph G is said to be an even vertex equitable even labeling if there exists a vertex labeling f : V (G) → A that induces an edge labeling f ∗ defined by f ∗ (uv) = f (u) + f (v) for all edges uv such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the induced edge labels are 2, 4, · · · , 2q, where vf (a) be the number of vertices v with f (v) = a for a ∈ A. A graph that admits even vertex equitable even labeling is called an even vertex equitable even graph. Definition 1.2 The corona G1 ⊙ G2 of two graphs G1 (p1 , q1 ) and G2 (p2 , q2 ) is defined as the graph obtained by taking one copy of G1 and p1 copies of G2 and joining the ith vertex of G1 with an edge to every vertex in the ith copy of G2 . Definition 1.3 The graph G = Pn (Qm ) is defined as isomorphic Quadrilateral snake attached with each vertex of path Pn , m is the number of C4 attached in the Quadrilateral snake. Definition 1.4 The super subdivision graph S ∗ (G) is obtained from G by replacing every edge e of G by a complete bipartite graph K2,m , (m ≥ 2) in such a way that the ends of e are merged with the two vertices of the 2-vertices part of K2,m after removing the edge e from G. Definition 1.5 The graph Pn × P2 is called a ladder graph Ln . Definition 1.6 The bistar Bn,n is the graph obtained by attaching the apex vertices of two copies of K1,n by an edge. Definition 1.7 For a simple connected graph G the square of graph G is denoted by G2 and defined as the graph 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. ′
Definition 1.8 For a graph G the splitting graph S (G) of a graph G is obtained ′ by adding a new vertex v corresponding to each vertex v of G such that N (v) = ′ N (v ).
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Main Results
Theorem 2.1 The graph Pn ⊙ mK1 is an even vertex equitable even graph for n ≡ 0 (mod 2). S Proof: Let G = Pn ⊙ mK1 . Let V (G) = {uiS: 1 ≤ i ≤ n} {uij : 1 ≤ i ≤ n; 1 ≤ j ≤ m} and E(G) = {ui ui+1 : 1 ≤ i ≤ n − 1} {ui uij : 1 ≤ i ≤ n, 1 ≤ j ≤ m}. Then, G is of order mn + n and size mn + n − 1. Define f : V (G) → A = {0, 2, · · · , mn + n} as follows: For 1 ≤ i ≤ n, ( (m + 1)(i − 1) if i is odd f (ui ) = (m + 1)i if i is even ; ( (m + 1)(i − 1) + 2j if i is odd and 1 ≤ j ≤ m f (uij ) = (m + 1)(i − 2) + 2j if i is even and 1 ≤ j ≤ m. It can be verified that the induced edge labels of Pn ⊙mK1 are 2, 4, · · · , 2mn+2n−2 and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Thus, Pn ⊙ mK1 is an even vertex equitable even graph. Example 2.2 An even vertex equitable even labeling of P4 ⊙ 4K1 is shown in Figure 2.1. 0
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Figure 2.1 Theorem 2.3 The graph Pn (Qm ) is an even vertex equitable even graph. S Proof: Let G = Pn (Qm ). Let V (G) = {ui : 1 ≤ i ≤ n}S {uij , vij , wij : 1 ≤ i ≤ n; 1S≤ j ≤ m} and E(G) = {ui ui+1 : 1 S ≤ i ≤ n − 1} {ui vi1 , ui wi1 : 1 ≤ i ≤ n} {vij uij , wij uij : 1 ≤ i ≤ n; 1 ≤ j ≤ m} {uij vij+1 , uij wij+1 : 1 ≤ i ≤ n; 1 ≤ j ≤ m − 1}. Then, G is of order ( 3mn + n and size 4mn + n − 1. 0, 2, · · · , 4mn + n if 4mn + n − 1 is odd Define f : V (G) → A = 0, 2, · · · , 4mn + n − 1 if 4mn + n − 1 is even as follows: ( (4m + 1)(i − 1) if i is odd and 1 ≤ i ≤ n f (ui ) = (4m + 1)i if i is even and 1 ≤ i ≤ n ; For 1 ≤ i ≤ n and 1 ≤ j ≤ m, ( (4m + 1)(i − 1) + 4j if i is odd f (uij ) = (4m + 1)i − 4j if i is even ; ( (4m + 1)(i − 1) + 4(j − 1) + 2 if i is odd f (vij ) = (4m + 1)i − 4j if i is even ;
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(4m + 1)(i − 1) + 4j if i is odd (4m + 1)i − 4j + 2 if i is even. It can be verified that the induced edge labels of Pn (Qm ) are 2, 4, · · · , 8nm + 2n − 2 and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Thus, Pn (Qm ) is an even vertex equitable even graph. f (wij ) =
Example 2.4 An even vertex equitable even labeling of P3 (Q3 ) is shown in Figure 2.2. 0
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Figure 2.2 Theorem 2.5 The super subdivision graph S ∗ (Pn ⊙ K1 ) is an even vertex equitable even graph. Proof: Let u1 , u2 , · · · , un , v1 , v2 , · · · , vn be the vertices of Pn ⊙ K1 . The super subdivision graph S ∗ (Pn ⊙ K1 ) is obtained by replacing each edge of Pn ⊙ K1 is replaced by a completeSbipartite graph K2,m . Let V (S ∗ (Pn ⊙ K1 )) = {ui , vi , vij : 1 ≤ i ≤ n, 1 ≤ j ≤ m} {uij : 1 ≤ iS≤ n − 1, 1 ≤ j ≤ m} and E(S ∗ (Pn ⊙ K1 )) = {ui vij , vij vi : 1 ≤ i ≤ n, 1 ≤ j ≤ m} {ui uij , uij ui+1 : 1 ≤ i ≤ n − 1, 1 ≤ j ≤ m}. Then S ∗ (Pn ⊙ K1 ) is of order 2n + 2mn − m and size 4mn − 2m. Define f : V (S ∗ (Pn ⊙ (K1 )) → {0, 2, · · · , 4mn − 2m} as follows: 2m(2i − 1) if i is odd and 1 ≤ i ≤ n f (ui ) = 4m(i − 1) if i is even and 1 ≤ i ≤ n ; ( 4m(i − 1) if i is odd and 1 ≤ i ≤ n f (vi ) = 2m(2i − 1) if i is even and 1 ≤ i ≤ n ; For 1 ≤ j ≤ m, ( 2m(2i − 1) + 2j if i is odd and 1 ≤ i ≤ n − 1 f (uij ) = 4mi + 2j if i is even and 1 ≤ i ≤ n − 1 ; if i = 1 2j f (vij ) = 2m(2i − 1) − 2(j − 1) if i is even and 2 ≤ i ≤ n 4m(i − 1) − 2(j − 1) if i is odd and 2 ≤ i ≤ n. It can be verified that the induced edge labels of S ∗ (Pn ⊙ K1 ) are 2, 4, · · · , 8mn − 4m
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and |vf (a)−vf (b)| ≤ 1 for all a, b ∈ A. Thus, S ∗ (Pn ⊙K1 ) is an even vertex equitable even graph. Example 2.6 An even vertex equitable even labeling of S ∗ (P3 ⊙ K1 ) is shown in Figure 2.3. 8
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Figure 2.3 Theorem 2.7 The super subdivision graph S ∗ (Ln ) is an even vertex equitable even graph. Proof: Let u1 , u2 , · · · , un , v1 , v2 , · · · , vn be the vertices of Ln . The super subdivision graph S ∗ (Ln ) is obtained by replacing each edge of Ln is replaced by a complete bipartite graph K2,m . Let V (S ∗ (Ln )) = {ui , vi , wij : 1 ≤ i ≤ n, 1 ≤ j ≤ S m} {uij , vij : 1 S ≤ i ≤ n − 1, 1 ≤ j ≤ m} and E(S ∗ (Ln )) = {ui wij , vi wij : 1 ≤ i ≤ n, 1 ≤ j ≤ m} {ui uij , uij ui+1 , vi vij , vij vi+1 : 1 ≤ i ≤ n − 1, 1 ≤ j ≤ m}. Then S ∗ (Ln ) is of order 2nm + 2n − m and size 6mn − 4m. Define f : V (S ∗ (Ln ))(→ {0, 2, · · · , 6mn − 4m} as follows: 6m(i − 1) if i is odd and 1 ≤ i ≤ n f (ui ) = 2m(3i − 2) if i is even and 1 ≤ i ≤ n ; ( 2m(3i − 2) if i is odd and 1 ≤ i ≤ n f (vi ) = 6m(i − 1) if i is even and 1 ≤ i ≤ n ; For 1 ≤ j ≤ m, ( 6mi + 2j if i is odd and 1 ≤ i ≤ n − 1 f (uij ) = 2m(3i − 2) + 2j if i is even and 1 ≤ i ≤ n − 1 ; ( 2m(3i − 2) + 2j if i is odd and 1 ≤ i ≤ n − 1 f (vij ) = 6mi + 2j if i is even and 1 ≤ i ≤ n − 1 ; ( 2j if i = 1 f (wij ) = 6m(i − 1) − 2(j − 1) if 2 ≤ i ≤ n. It can be verified that the induced edge labels of S ∗ (Ln ) are 2, 4, · · · , 12mn − 8m and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Thus, S ∗ (Ln ) is an even vertex equitable even graph. Example 2.8 An even vertex equitable even labeling of S ∗ (L3 ) is shown in Figure 2.4.
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Figure 2.4 Theorem 2.9 The super subdivision graph S ∗ (Bn,n ) is an even vertex equitable even graph. Proof: Let u, u1 , u2 , · · · , un , v, v1 , v2 , · · · , vn be the vertices of the bistar Bn,n . The super subdivision graph S ∗ (Bn,n ) is obtained by replacing each edge of Bn,n is replaced by a complete S bipartite graph K2,m . Let V (S ∗ (Bn,n )) = {ui , vi , wj , uij , vij : 1 ≤ i ≤ n, 1 ≤ j ≤ m} {u, v} and E(S ∗ (Bn,n )) = {ui uij , uij u, uwj , wj v, vvij , vi vij : 1 ≤ i ≤ n, 1 ≤ j ≤ m}. Then S ∗ (Bn,n ) is of order 2mn+m+2n+2 and size 4mn+2m. Define f : V (S ∗ (Bn,n )) → {0, 2, · · · , 4mn + 2m} as follows: f (v) = 2m; f (u) = 4nm; f (vi ) = 4m(i − 1), 1 ≤ i ≤ n; f (ui ) = 2m(2i + 1), 1 ≤ i ≤ n; For 1 ≤ j ≤ m, f (wj ) = 4mn ( − 2(j − 1); 2j if i = 1 f (vij ) = 4m(i − 1) − 2(j − 1) if 2 ≤ i ≤ n ; f (uij ) = 2m(2i + 1) − 2(j − 1), 1 ≤ i ≤ n. It can be verified that the induced edge labels of S ∗ (Bn,n ) are 2, 4, · · · , 8mn + 4m and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Thus, S ∗ (Bn,n ) is an even vertex equitable even graph. Example 2.10 An even vertex equitable even labeling of S ∗ (B2,2 ) is shown in Figure 2.5. 2
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Figure 2.5 2 is an even vertex equitable even graph. Theorem 2.11 The graph Bn,n
Proof: Let u, u1 , u2 , · · · , un , v, v1 , v2 , · · · , vn be the vertices of the S bistar Bn,n . 2 with V (G) = {u, v, u , v : 1 ≤ i ≤ n} and E(G) = {uv} {uu , vv , uv , vu : Let G = Bn,n i i i i i i 1 ≤ i ≤ n}. Then V (G) = 2n + 2 and E(G) = 4n + 1. Define f : V (G) → {0, 2, · · · , 4n + 2} as follows: f (u) = 0; f (v) = 4n + 2; f (ui ) = 2i, 1 ≤ i ≤ n; f (vi ) = 2n + 2i, 1 ≤ i ≤ n. 2 It can be verified that the induced edge labels of Bn,n are 2, 4, · · · , 8n + 2 and 2 |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Thus, Bn,n is an even vertex equitable even graph. 2 is shown in Figure Example 2.12 An even vertex equitable even labeling of B4,4 2.6.
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Theorem 2.13 The graph S (Bn,n ) is an even vertex equitable even graph. Proof: Let u, u1 , u2 , · · · , un , v, v1 , v2 , · · · , vn be the vertices of the bistar Bn,n . S ′ ′ ′ ′ ′ Let G = S (BS ) with V (G) = {u , v , u , v : 1 ≤ i ≤ n} {u, v, u , v } and E(G) = n,n i i i i ′ ′ ′ ′ ′ ′ {uv, uv , vu } {uui , uui , u ui , vvi , vvi , v vi , : 1 ≤ i ≤ n}. Then V (G) = 4n + 4 and E(G) = 6n + 3. Define f : V (G) → {0, 2, · · · , 6n + 4} as follows: f (u) = 0; f (v) = 6n + 4; ′ f (u ) = 2n + 2; ′ f (v ) = 4n + 2; f (ui ) = 2n + 2i, 1 ≤ i ≤ n; ′ f (ui ) = 2i, 1 ≤ i ≤ n; f (vi ) = 2n + 2 + 2i, 1 ≤ i ≤ n;
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f (vi ) = 4n + 2 + 2i, 1 ≤ i ≤ n. ′ It can be verified that the induced edge labels of S (Bn,n ) are 2, 4, · · · , 12n + 6 and ′ |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Thus, S (Bn,n ) is an even vertex equitable even graph. ′
Example 2.14 An even vertex equitable even labeling of S (B5,5 ) is shown in Figure 2.7. 2
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References [1] J.A. Gallian, A dyamic durvey of graph labeling, The Electronic J. Combin., 18 (2015), # DS6, 1-389. [2] F. Harary, Graph Theory, Addison-wesley, Reading, Mass, 1972. [3] A. Lourdusamy, J.S. Mary and F. Patrick, Even vertex equitable even labeling, Sciencia Acta Xaveriana, 7(1) (2016), 37-46. [4] A. Lourdusamy, J.S. Mary and F. Patrick, Even vertex equitable even labeling for path related graphs, Sciencia Acta Xaveriana, 7(1) (2016), 47-56. [5] A. Lourdusamy, J.S. Mary and F. Patrick, Some results on even vertex equitable even labeling, (Submitted for publication). [6] A. Lourdusamy and F. Patrick, Even vertex equitable even labeling for corona and Tp -tree related graphs, (Accepted in Utilitas Mathematica). [7] A. Lourdusamy and F. Patrick, Even vertex equitable even labeling for cycle related graphs, (Submitted for publication). [8] A. Maheswari, A study on graphs labeling: Equitable labeling and related concepts, PhD Thesis, Manonmaniam Sundaranar University, India, (2013).
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c Copyright 2016 A. Lourdusamy and F. Patrick. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
