Magic Graphoidal on Class of Trees A. Nellai Murugan

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1 Bulletin of Mathematical Sciences and Applications Online: ISSN: , Vol. 9, pp doi: / SciPress Ltd., Switzerland Magic Graphoidal on Class of Trees A. Nellai Murugan Department of Mathematics, V.O.Chidambaram College, Tuticorin, Tamilnadu. Keywords: Graphoidal Cover, Magic Graphoidal, Graphoidal Constant. Abstract: B.D.Acharya and E. Sampathkumar [1] defined Graphoidal cover as partition of edge set of a graph G into internally disjoint paths (not necessarily open). The minimum cardinality of such cover is known as graphoidal covering number of G. Let G = {V, E} be a graph and let ψ be a graphoidal cover of G. Define f: V E {1, 2,, p+ q} such that for every path P = (v0,v1,v2, vn) in ψ with f * (P) = f(v0) + f(vn) + (v1i 1v ) = k, a constant, where f * is the induced labeling on ψ. Then, we say that G admits ψ - magic graphoidal total labeling of G. A graph G is called magic graphoidal if there exists a minimum graphoidal cover ψ of G such that G admits ψ - magic graphoidal total labeling. In this paper, we proved that [Pn;S1], [Pn;S2], T(n), Pm K1,3, Pm 2K1 and K1,n 2 K1 are magic graphoidal 1. INTRODUCTION By a graph, we mean a finite simple and undirected graph. The vertex set and edge set of a graph G are denoted by V(G) and E(G) respectively. Terms and notations not used here are as in [3] Definition : Let S 1 = (v 0, v 1 ) be a star and let [P n ; S 1 ] be the graph obtained from n copies of S 1 and the path P n = (u 1, u 2,, u n ) by joining u j with the vertex v 0 of the j th copy of S 1 by means of an edge, for 1 j n Definition : Let S 2 = (v 0,v 1,v 2 ) be a star and let [P n ; S 2 ] be the graph obtained from n copies of S 2 and the path P n = (u 1, u 2,, u n ) by joining u j with the vertex v 0 of the j th copy of S 2 by means of an edge, for 1 j n Defintion : Let T be any Tree. Denote the tree, obtained from T by considering two copies of T by adding an edge between them, by T (2) and in general the graph obtained from T (n- 1) and T by adding an edge between them is denoted by T (n) Result [11] : For a Tree T, γ(t) = n-1 where n is the number of pendent vertices of G. 2. PRELIMINARIES Let G = {V, E} be a graph with p vertices and q edges. A graphoidal cover ψ of G is a collection of (open) paths such that (i) every edge is in exactly one path of ψ (ii) every vertex is an interval vertex of atmost one path in ψ. We define γ(g) = where ζ is the collec tion of graphoidal covers ψ of G and γ is graphoidal covering number of G. SciPress applies the CC-BY 4.0 license to works we publish:

2 34 Volume 9 Let ψ be a graphoidal cover of G. Then we say that G admits ψ - magic graphoidal total i labeling of G if there exists a bijection f: V E {1, 2,,p+ q} such that for every path P =(v 0 v 1 v 2 v n ) in ψ, then, f * (P) = f(v 0 ) + f(v n ) + (v1i 1v ) = k, a constant, where f * is the induced labeling of ψ. A graph G is called magic graphoidal if there exists a minimum graphoidal cover ψ of G such that G admits ψ - magic graphoidal total labeling. 3. Magical Graphoidal on Trees 3.1. Theorem : [P n ; S 1 ], (n - even) is magic graphoidal. Proof: Let G = [P n ; S 1 ] Let V(G) = {u i, v i, w i : 1 i n} and E(G) = { [(u i v i ) (v i w i ) : 1 i n ] [(u i u i+1 ) : 1 i n-1]} Define f : V E {1, 2,., p+q} by f(w 1 ) = 1 f(w 1 v 1 ) = 2 f(v 1 u 1 ) = 3 f(u i+1 ) = 6 + i 1 i n -2 f(w i+1 ) = 6n i 1 i n -1 f(v i+1 w i+1 ) = 4n i 1 i n-1 f(u i+1 v i+1 ) = 3n + 3 2i 1 i (n\2)-1 Let ψ = {P 1 = (w 1,v 1,u 1,u 2,v 2,w 2 ), P 2 = (u i,u i+1,v i+1,w i+1 ) : 2 i n 1} f * [P 1 ] = f(w 1 ) + f(w 2 ) + f(w 1 v 1 ) + f(v 1 u 1 ) + f(u 1 u 2 ) + f(u 2 v 2 ) + f(v 2 w 2 ) For 2 i (n\2) 1, f * [P 2 ] = f(u i ) + f(w i+1 ) + f(u i u i+1 ) + f(u i+1 v i+1 ) + f(v i+1 w i+1 ) = 6 + i 1 + 6n i +(3n\2)+ 4 + i + 3n + 3 2i + 4n i = 13n +(3n\2) (B) For (n\2) i n 1,

3 Bulletin of Mathematical Sciences and Applications Vol f * [P 2 ] = f(u i ) + f(w i+1 ) + f(u i u i+1 ) + f(u i+1 v i+1 ) + f(v i+1 w i+1 ) = 6+i 1+6n i + n i (n\2)+1+3n+4 2(i+1 (n\2))+ 4n +1 + i = 13n +(3n\2) (C) From (A), (B) and (C), we conclude that G admits ψ - magic graphoidal total labeling. Hence, [P n ; S 1 ], (n - even) is magic graphoidal. For example, consider the graph [P 6 ; S 1 ] shown in figure 1. Figure 1 [P6 ; S1] Clearly, ψ = {(w 1,v 1,u 1,u 2,v 2,w 2 ), (u 2,u 3,v 3,w 3 ), (u 3,u 4,v 4,w 4 ), (u 4,u 5,v 5,w 5 ), (u 5,u 6,v 6,w 6 )} is a minimum graphoidal cover and [P 6 ; S 1 ] is magic graphoidal. Here the constant K = Theorem : [P n ; S 1 ], (n - odd) is magic graphoidal. Proof: Let G = [P n ; S 1 ] Let V(G) = {u i, v i, w i : 1 i n} and E(G) = { [ (u i v i ) (v i w i ) : 1 i n ] [(u i u i+1 ) : 1 i n-1]} Define f : V E {1, 2,., p+q} by f(w 1 ) = 1 f(w 1 v 1 ) = 2 f(v 1 u 1 ) = 3 f(u i+1 ) = 5 + 2i 1 i n-1)\2

4 36 Volume 9 Let ψ = {P 1 = (w 1,v 1,u 1,u 2,v 2,w 2 ), P 2 = (u i,u i+1,v i+1,w i+1 ) : 2 i n 1} f * [P 1 ] = f(w 1 ) + f(w 2 ) + f(w 1 v 1 ) + f(v 1 u 1 ) + f(u 1 u 2 ) + f(u 2 v 2 ) + f(v 2 w 2 ) For 2 i < From (A), (B) and (C), we conclude that ψ is minimum magic graphoidal cover Hence, [P n ; S 1 ], (n - even) is magic graphoidal. For example, consider the graph [P 7 ; S 1 ] shown in figure 2. Figure 2 [P 7 ; S 1 ] Clearly, ψ = {(w 1,v 1,u 1,u 2,v 2,w 2 ), (u 2,u 3,v 3,w 3 ), (u 3,u 4,v 4,w 4 ), (u 4,u 5,v 5,w 5 ), (u 5,u 6,v 6,w 6 ), (u 6,u 7,v 7,w 7 )} is a minimum graphoidal cover and [P 7 ; S 1 ] is magic graphoidal. Here the constant K = Theorem : [P n ; S 2 ] is magic graphoidal. Proof: Let G = [P n ; S 2 ] V(G) = {(u i, v i ) : 1 i n}

5 Bulletin of Mathematical Sciences and Applications Vol E(G) = { [ (u i u i+1 ) : 1 i n-1] [(u i v i ) : 1 i n] [(v i v i1 ) (v i v i2 ) : 1 i n] } Define f : V E {1, 2,., p+q} by f(v 1 ) = 1 f(u 1 v 1 ) = 2 f (u 1 ) = 3 f(u i+1 ) = 3 + i 1 i n -2 f(u n )= p + q f(v i1 )= n i 1 i n f(v i+1 ) = 8n 1 i 1 i n -1 f(v i2 ) = 7n i 1 i n f(v i v i2 ) = 5n 1 + i 1 i n f(u i+1 v i+1 ) = 4n + i 1 i n 1 f(u i u i+1 ) = 4n + 1 i 1 i n 1 f(v i v i1 ) = 3n + 2 i 1 i n Let ψ = {P 1 = [(v i1,v i,v i2 ): 1 i n], P 2 =(v 1,u 1,u 2,v 2 ), P 3 = [(u i,u i+1,v i+1 ) : 2 i n 1]} For 1 i n, f * [P 1 ] = f(v i1 ) + f(v i2 ) + f(v i1 v i ) + f(v i v i2 ) = n +1 + i + 7n i + 3n + 2 i + 5n 1 + i = 16n (A) f * [P 2 ] = f(v 1 ) + f(v 2 ) + f(v 1 u 1 ) + f(u 1 u 2 ) + f(u 2 v 2 ) = 1 + 8n n + 4n + 1 = 16 n (B) For 2 i n 1, f * [P 3 ] = f(u i ) + f(v i+1 ) + f(u i u i+1 ) + f(u i+1 v i+1 ) = 3 + i 1 + 8n 1 i + 4n + 1 i + 4n + i = 16 n (C) From (A), (B) and (C), we conclude that G admits ψ - magic graphoidal total labeling. Hence, [P n ; S 2 ] is magic graphoidal. For example, consider the graphs [P 3 ; S 2 ] and [P 4 ; S 2 ] shown in figure 3.1 and 3.2. Figure 3.1 [P 3 ; S 2 ] Clearly, ψ = {(v 11,v 1,v 12 ), (v 21,v 2,v 22 ), (v 31,v 3,v 32 ), (v 1,u 1,u 2,v 2 ), (u 2,u 3,v 3 )} is a minimum graphoidal cover and [P 3 ; S 2 ] is magic graphoidal. Here the constant K = 50

6 38 Volume 9 Figure 3.2 [P 4 ; S 2 ] Clearly, ψ = {(v 11,v 1,v 12 ), (v 21,v 2,v 22 ), (v 31,v 3,v 32 ), (v 41,v 4,v 42 ), (v 1,u 1,u 2,v 2 ), (u 2,u 3,v 3 ), (u 3,u 4,v 4 )} is a minimum graphoidal cover and [P 4 ; S 2 ] is magic graphoidal. Here the constant K = Theorem : For a Tree, T (n) is magic graphoidal. Proof : Let T (n) be a graph such that V[T (n) ] = {u i1, u i2, u i3, u i4, u i5 : 1 i n } and E[T (n) ] = {[(u i1 u i2 ), (u i2 u i3 ), (u i3 u i4 ), ( u i4 u i5 ): 1 i n] [( u in u i+1n ) : 1 i n-1)]} Define f : V E {1, 2,., p+q} by f(u i1 ) = i 1 i n f(u i3 ) = n + 1 f(u i3 u i5 ) = n + 2 f(u i1 u i2 ) = 2n +3 i 1 i n f(u i+2,5 u i+3,5 ) = 5n 3 2i 1 i n 3 f(u n+1-i,5 u n-i,5 ) = 4n 1 + i n 2 i n 1 f(u n+1-i,2 u n+1-i,3 ) = 5n 2 + i 1 i n f(u n+1-i,3 u n+1-i,4 ) = 6n 2 + i 1 i n f(u n+1-i,3 u n+1-i,5 ) = 7n 2 + i 1 i n 1 f(u i4 ) = 8n 2 +2(i-1) 1 i n f(u i+1,3 ) = 8n 1 +2(i-1) 1 i n 1 Let ψ = {P 1 = (u i1,u i2,u i3,u i4 ), P 2 =(u i3,u i5,u 25,u 23 ), P 3 = [(u i5,u i+15, u i+13 ) : 2 i n 1]} f * [P 1 ] = f(u i1 ) + f(u i4 ) + f(u i1 u 12 ) + f(u i2 u i3 ) + f(u i3 u i4 ) = i+8n 2+2(i-1)+2n+3 i+5n 2 + (n + 1 i ) + 6n 2 + (n + 1 i ) = 23n (A) f * [P 2 ] = f(u i5 ) + f(u i+1,3 ) + f(u 5 u i+1,3 ) + f(u i+1,5 u i+1,3 ) = 2n+2+i 1+8n 1 + 2( i 1) + 5n 3 2 (i 2) + 7n 2+ n i = 23n (B) f * [P 3 ] = f(u i3 ) + f(u 23 ) + f(u i3 u 15 ) + f(u 15 u 25 ) + f(u 25 u 23 ) = n n 1 + 2(i 1) + 5n 3 2(i 2) + 7n 2 + n i = 23n (C) From (A), (B) and (C), we conclude that G admits ψ - magic graphoidal total labeling. Hence, T (n) is magic graphoidal.

7 Bulletin of Mathematical Sciences and Applications Vol For example, consider the graph T (3) shown in figure 4. Figure 4 T (3) Clearly, ψ = {(u 11,u 12,u 13,u 14 ), (u 21,u 22,u 23,u 24 ), (u 31,u 32,u 33,u 34 ), (u 13,u 15,u 25,u 23 ), (u 25,u 35,u 33 )} is a minimum graphoidal cover and T (3) is magic graphoidal. Here the constant K = Theorem : The graph Double Crowned Star K 1,n 2K 1 is magic graphoidal. Proof : Let G = K 1,n 2K 1 V(G) = {u, [u i : 1 i n], [(u i1,u i2 ): 1 i n] } and E(G) = { [(uu i ): 1 i n] [ (u i u i1 ) (u i u i2 ) : 1 i n ] } Define f : V E {1, 2,., p+q} by f(u) = 2n + 3 f(u i1 ) = i 1 i n f(u 1 ) = n +1 f(u 2 ) = n +2 f(u n+1-i,2 ) = n +2 + i 1 i n f(u i u i,1 ) = 2n +3 + i 1 i n f(uu 1 ) = 3n +4 f(u 2+i ) = 3n +4 + i 1 i n 2 f(uu n+1-i ) = 4n +2 + i 1 i n 1 f(u n+1-i u n+1-i, 2 ) = 5n +1 + i 1 i n Let ψ = {P 1 = (u 1,u,u 2 ), P 2 = [(u,u i ) : 3 i n], P 3 = [(u i1,u i,u i2 ) : 1 i n]} f * [P 1 ] = f(u 1 ) + f(u 2 ) + f(u 1 u) + f(uu 2 ) = n n n n + 1 = 10n (A) For 3 i n, f * [P 2 ] = f(u) + f(u i ) + f(uu i ) = 2n n i 2 + 4n n + 1 i = 10n (B) For 1 i n,

8 40 Volume 9 f * [P 3 ] = f(u i1 ) + f(u i2 ) + f(u i1 u i ) + f(u i u i2 ) = i + n n + 1 i + 2n i + 5n n + 1 i = 10n (C) From (A), (B) and (C), we conclude that G admits ψ - magic graphoidal total labeling. Hence, Double Crowned Star K 1,n 2K 1 is magic graphoidal. For example, consider the graph K 1,5 2K 1 shown in figure 5. Figure 5. K 1,5 2K 1 Clearly, ψ = {(u 11,u 1,u 12 ), (u 21,u 2,u 22 ), (u 31,u 3,u 32 ), (u 41,u 4,u 42 ), (u 51,u 5,u 52 ), (u 1,u,v,u 2 ), (u,u 3 ), (u,u 4 ), (u,u 5 )} is a minimum graphoidal cover and K 1,5 2K 1 is magic graphoidal. Here the constant K = Theorem : P m 2K 1 is magical graphoidal. Proof : Let G = P m 2K 1 V(G) = { (u i : 1 i m), (u ij : 1 i m, 1 j 2) } and E(G) = { [(u i u i+1 ) : 1 i m-1] [(u i u ij ) : 1 i m, 1 j 2] } Let ψ = {P 1 =[(u i1,u i,u i2 ) : 1 i m], P 2 = [(u i,u i+1 ) : 1 i m-1]} Define f : V {0, 1, 2,, 6m-1} by f(u i u i1 ) = i 1 i m f(u i u i2 ) = 2m+1 i 1 i m

9 Bulletin of Mathematical Sciences and Applications Vol f(u m+1-i u m-i ) = 4m + i 1 i m -1 f(u m+1-i, 2 ) = 5m 1 + i 1 i m Case (i) : when m is odd For 1 i m, f * [P 1 ] = f(u i1 ) + f(u i2 ) + f(u i1 u i ) + f(u i u i2 ) For 1 i m 1, i 1 mod 2 f * [P 2 ] = f(u i ) + f(u i+1 ) + f(u i u i+1 ) For 1 i m 1, i 0 mod 2 f * [P 2 ] = f(u i ) + f(u i+1 ) + f(u i u i+1 ) From (A), (B) and (C), we conclude that G admits ψ - magic graphoidal total labeling. Hence, P m 2K 1 (m-odd) is magic graphoidal. For example, consider the graph P 5 2K 1 shown in figure 6.1. Case (ii) : when m is even For 1 i m, f * [P 1 ] = f(u i1 ) + f(u i2 ) + f(u i1 u i ) + f(u i u i2 )

10 42 Volume 9 5m =+ i + 5m 1 + m + 1 i + i + 2m i 2 = 21m (A) For 1 i m -1, i 1 mod 2 f * [P 2 ] = f(u i ) + f(u i+1 ) + f(u i u i+1 ) = For 1 i m -1, i 0 mod 2 f * [P 2 ] = f(u i ) + f(u i+1 ) + f(u i u i+1 ) From (A), (B) and (C), we conclude that G admits ψ - magic graphoidal total labeling. Hence, P m 2K 1 (m-even) is magic graphoidal. For example, consider the graph P 4 2K 1 shown in figure 6.2. Figure 6.1 P 5 2K 1 Clearly, ψ = {(u 11,u 1,u 12 ), (u 21,u 2,u 22 ), (u 31,u 3,u 32 ), (u 41,u 4,u 42 ), (u 51,u 5,u 52 ), (u 1,u 2 ), (u 2,u 3 ), (u 3,u 4 ), (u 4,u 5 )} is a minimum graphoidal cover and P 4 2K 1 is magic graphoidal. Here the constant K = 53.

11 Bulletin of Mathematical Sciences and Applications Vol Figure 6.2 P 4 2K 1 Clearly, ψ = {(u 11,u 1,u 12 ),(u 21,u 2,u 22 ),(u 31,u 3,u 32 ),(u 41,u 4,u 42 ),(u 1,u 2 ),(u 2,u 3 ),(u 3,u 4 )} is a minimum graphoidal cover and P 4 2K 1 is magic graphoidal. Here the constant K = Theorem : P m K 1,3 is magic graphoidal. Proof : Let G = P m K 1,3 V(G) = {[(u i v i ): 1 i m], [(v ij ): 1 i m, 1 i 3]} E(G) = { [(u i u i+1 ): 1 i m-1] [ (v i v ij ) : 1 i m, 1 j 3 ] } with u i = u i3, 1 i m Let ψ = {P 1 = [(v i1,v i,v i2 ) : 1 i m], P 2 =[(v i,u i,u i+1 ): 1 i m 2], Define f : V E {1, 2,., 8m 1} by For 1 i m, P 3 =(v m-1,u m-1,u m,v m )} f(v m ) = 1 f(v i v i1 ) = i+1 1 i m f(u i v i ) = m+1+ i 1 i m 1 f(u m+1-i u m-i ) = 2m + i 1 i m 1 f(v m+1-i v m+1-i,2 ) = 3m 1+ i 1 i m f(v i1 ) = 4m 1 + i 1 i m f(v i ) = 5m 1 + i 1 i m 1 f(u m v m ) = 6m 1 f(u m-i ) = 6m + i 1 i m 2 f(v m+1-i, 2 ) = 7m 2 + i 1 i m f * [P 1 ] = f(v i1 ) + f(v i2 ) + f(v i1 v i ) + f(v i v i2 ) = 4m 1+ i + 7m 2 + m + 1 i + i m 1 + m + 1 i = 16m (A) For 1 i m 2, f * [P 2 ] = f(v i ) + f(u i+1 ) + f(v i u i ) + f(u i u i+1 ) = 5m 1 + i+ 6m + (m i 1) + m i + 2m + m i = 16m (B)

12 44 Volume 9 f * [P 3 ] = f(v m-1 ) + f(v m ) + f(v m-1 u m-1 ) + f(u m-1 u m ) + f(u m v m ) = 6m m + 2m m 1 = 16m (C) From (A), (B) and (C), we conclude that G admits ψ - magic graphoidal total labeling. Hence, P m K 1,3 is magic graphoidal. For example, consider the graph P 4 K 1,3 shown in figure 7. Figure 7. P 4 K 1,3 Clearly, ψ = {(v 11,v 1,v 12 ), (v 21,v 2,v 22 ), (v 31,v 3,v 32 ), (v 41,v 4,v 42 ), (v 1,u 1,u 2 ), (v 1,u 2,u 3 ), (v 3,u 3,u 4,v 4 )} is magic graphoidal. Here the constant K = 63. REFERENCES [1] B.D.Acarya and E. Sampath Kumar, Graphoidal covers and Graphoidal covering number of a Graph, Indian J.Pure Appl. Math. 18(10).(1987) [2] J.A. Gallian, A Dynamic Survey of graph labeling, The Electronic journal of Coimbinotorics 6 (2001) # DS6. [3] F. Harary, Graph Theory, Addition - Wesley publishing company Inc, USA, [4] A. Nellai Murugan, Magic graphoidal on Sm,n Outreach, A Multidisciplinary Refreed Journal, Vol. 4 ( ), [5] A. Nellai Murugan and A. Nagarajan, Magic Graphoidal on Special type of Unicyclic Graphs, Scientia Magna, Vol.7, (2011), No. 1, [6] A. Nellai Murugan and A.Nagarajan Magic graphidal of path related graphs Advance Journal of Physical sciences, Access International Journals, Vol 1(2) pp 14-25, December 2012 [7] A. Nellai Murugan, Magic graphiadal on join of two graphs, International Journal of mathematical combinatorics. Vol 4(2012), [8] A. Nellai Murugan, Magic graphiadal on product graphs, International journal of innovative research & development Vol 1(8),2012, [9] C. Packiam and S. Arumugam, On the Graphoidal covering number of a Graph. Indian J.Pure Appl. Math. 20 (1989),

13 Volume / Magic Graphoidal on Class of Trees /