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2. 3-Equitable Labeling for Some Star and Bistar Related Graphs

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2. 3-Equitable Labeling for Some Star and Bistar Related Graphs INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 3 3-Equitable Labeling for Some Star and Bistar Related Graphs S.K. Vaidya and N.H. Shah Abstract—In this paper we prove that the splitting graphs of The concept of 3-equitable labeling was introduced by K B 1,n and n,n are 3-equitable graphs. We also show that the Cahit [1] and he proved that an Eulerian graph with number B shadow graph of n,n is a 3-equitable graph. Further we prove of edges congruent to 3(mod 6) is not 3-equitable. In the that square graph of Bn,n is 3-equitable for n ≡ 0(mod 3) and n ≡ 1(mod 3) and not 3-equitable for n ≡ 2(mod 3). same paper he proved that Cn is 3-equitable if and only if n 6≡ 3(mod 6) and all caterpillars are 3-equitable. Cahit Index Terms—3-equitable labeling, star, bistar, splitting graph, [1] claimed to prove that W is 3-equitable if and only if shadow graph, square graph. n n 6≡ 3(mod 6) but Youssef [8] proved that Wn is 3-equitable MSC 2010 Codes - 05C78. for all n ≥ 4. The 3-equitable labeling in the context of vertex duplication is discussed by Vaidya et al. [5] while same authors in [6] have investigated 3-equitable labling for I. INTRODUCTION some shell related graphs. Vaidya et al. [7] have discussed 3- N this paper we consider simple, finite, connected and equitablity of graphs in the context of some graph operations and proved that the shadow and middle graph of cycle C , undirected graph G = (V (G),E(G)) with order p and n I P size q. For all standard terminology and notations we follow path n are 3-equitable. West [9]. We will give brief summary of definitions which are useful for the present investigations. Generally there are three types of problems that can be considered in this area. Definition 1.1 : If the vertices of the graph are assigned values subject to certain condition(s) then is known as graph labeling. 1) How 3-equitability is affected under various graph op- A detailed study on applications of graph labeling is erations? reported in Bloom and Golomb [3]. According to Beineke 2) Construct new families of 3-equitable graph by investi- and Hegde [2] graph labeling serves as a frontier between gating suitable labeling. number theory and structure of graphs. For an extensive 3) Given a graph theoretic property P, characterize the class survey on graph labeling and bibliographic references we of graphs with property P that are 3-equitable. refer to Gallian [4]. The problems of second type are largely discussed while the problems of first and third types are rarely discussed and Definition 1.2 : Let G = (V (G),E(G)) be a graph. A they are of great importance also. The present work is aimed mapping f : V (G) →{0,1,2} is called ternary vertex labeling to discuss the problems of first kind. of G and f(v) is called the label of the vertex v of G under f. Definition 1.4 : The splitting graph of a graph G is obtained ′ ′ For an edge e = uv, the induced edge labeling by adding to each vertex v a new vertex v such that v f ∗ : E(G) → {0, 1, 2} is given by f ∗(e)= |f(u) − f(v)|. Let is adjacent to every vertex that is adjacent to v in G, i.e. N v N v′ . The resultant graph is denoted by S′ G . vf (0), vf (1) and vf (2) be the number of vertices of G having ( )= ( ) ( ) labels 0,1 and 2 respectively under f and let ef (0),ef (1) Definition 1.5: The shadow graph D2(G) of a connected and ef (2) be the number of edges having labels 0,1 and 2 ′ respectively under f ∗. 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 ′′ ′′ Definition 1.3 : A ternary vertex labeling of a graph G corresponding vertex u in G . is called a 3-equitable labeling if |vf (i) − vf (j)| ≤ 1 and Definition 1.6: For a simple connected graph G the square of |ef (i) − ef (j)| ≤ 1 for all 0 ≤ i,j ≤ 2. A graph G is 2 3-equitable if it admits 3-equitable labeling. graph G is denoted by G 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. Dr. S. K. Vaidya is a proffesor at the Department of Mathematics, Saurashtra University, Rajkot, Gujarat - 360005, INDIA (e-mail: [email protected]) N. H. Shah is a lecturer at Government Polytechnic, Rajkot, Gujarat - 360003, INDIA (e-mail: [email protected]) INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 4 II. MAIN RESULTS ef (0) = ef (1) = ef (2) = n Theorem 2.1 : S′ K is 3-equitable graph. ( 1,n) Thus in each case we have |vf (i) − vf (j)| ≤ 1 and Proof : Let v ,v ,v ,...,v be the pendant vertices and 1 2 3 n |ef (i) − ef (j)|≤ 1, for all 0 ≤ i,j ≤ 2. v be the apex vertex of K and u,u ,u ,u ,...,u are ′ 1,n 1 2 3 n Hence S (K1,n) is a 3-equitable graph. added vertices corresponding to v,v1,v2,v3,...,vn to obtain ′ ′ S (K1,n). Let G be the graph S (K1,n) then |V (G)| = 2n+2 ′ Illustration 2.2 : 3-equitable labeling of the graph S (K , ) and |E(G)| = 3n. To define f : V (G) → {0, 1, 2} we consider 1 7 is shown in Fig. 1. u following three cases. u 4 3 u u 1 5 Case 1: n ≡ 0(mod 3) 2 u u 1 1 6 u 1 1 1 7 f(v) = 2, 0 2 f(u) = 0, n f(v ) = 0; 1 ≤ i ≤ + 1 i 3 2 v n n f v 3 +1+i = 1; 1 ≤ i ≤ − 1 3 v v ¡ ¢ n 1 7 n v v f v 2 +i = 2; 1 ≤ i ≤ 0 2 v v v 6 2 3 3 3 4 5 ³ ´ n 0 0 1 2 2 f(u ) = 0; 1 ≤ i ≤ − 1 i 3 n f u n − = 1; 1 ≤ i ≤ + 2 3 1+i 3 ¡ ¢ n f u 2n i 0 3 +1+i = 2; 1 ≤ ≤ − 1 ³ ´ 3 u In view of the above labeling patten we have Figure 1 2n v (0) = v (1) = +1= v (2) + 1 ′ f f 3 f Theorem 2.3 : S (Bn,n) is 3-equitable graph. ef (0) = ef (1) = ef (2) = n Proof : Consider Bn,n with vertex set {u,v,ui,vi, 1 ≤ i ≤ n} ′ where ui,vi are pendant vertices. In order to obtain S (Bn,n), Case 2: n ≡ 1(mod 3) ′ ′ ′ ′ add u ,v ,ui,vi vertices corresponding to u,v,ui,vi where, 1 ≤ i ≤ n. If G = S′(B ) then |V (G)| = 4(n + 1) and Since n ≡ 1(mod 3), n = 3k + 1 some k ∈ N. n,n |E(G)| = 6n+3. To define f : V (G) → {0, 1, 2} we consider f(v) = 2, following four cases. f(u) = 0, Case 1: n = 2, 5 ′ ′ f(vi) = 0; 1 ≤ i ≤ k + 1 The graphs S (B2,2) and S (B5,5) are to be dealt separately f (vk+1+i) = 1; 1 ≤ i ≤ k − 1 and their 3-equitable labeling is shown is Fig. 2. and Fig. 3. ' ' ' ' f (v2k+i) = 2; 1 ≤ i ≤ k + 1 u1 u2 v1 v2 f(ui) = 0; 1 ≤ i ≤ k − 1 2 0 2 0 f (uk−1+i) = 1; 1 ≤ i ≤ k + 3 f (u2k+2+i) = 2; 1 ≤ i ≤ k − 1 u 0 1 v In view of the above labeling patten we have u v 2n + 1 1 u2 1 v2 v (0) = v (2) = = v (1) − 1 2 2 1 1 f f 3 f ef (0) = ef (1) = ef (2) = n Case 3: n ≡ 2(mod 3) 0 1 u' v' Since n ≡ 2(mod 3), n = 3k + 2 some k ∈ N. Figure 2 ' ' ' u3 ' ' v3 ' u2 u4 v2 v4 f(v) = 2, ' 2 ' ' 0 ' u1 2 0 u5 v1 1 0 v5 f(u) = 0, 2 0 2 0 f(vi) = 0; 1 ≤ i ≤ k + 1 f (vk+1+i) = 1; 1 ≤ i ≤ k u 0 1 v f (v2k+1+i) = 2; 1 ≤ i ≤ k + 1 u1 v5 f(ui) = 0; 1 ≤ i ≤ k 2 u2 u4 1 1 v2 v4 0 u3 v3 2 2 u5 v1 1 1 f (uk+i) = 1; 1 ≤ i ≤ k + 2 2 1 f (u2k+2+i) = 2; 1 ≤ i ≤ k In view of the above labeling patten we have 0 1 2(n + 1) u' v' vf (0) = vf (2) = vf (1) = ¹ 3 º Figure 3 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 1, 2012 5 Case 2: n ≡ 0(mod 3) In view of the above labeling patten we have Since n ≡ 0(mod 3), n = 3k some k ∈ N. vf (0) = vf (1) = 4(k + 1) = vf (2) f(u) = 0, ef (0) = ef (1) = ef (2) = 2n + 1 f(u′) = 0, f(v) = 1, Thus in each case we have |vf (i) − vf (j)| ≤ 1 and ′ f(v ) = 1, |ef (i) − ef (j)|≤ 1, for all 0 ≤ i,j ≤ 2.
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