Vertex Graceful Labeling-Some Path Related Graphs

Transcription

1 Internatonal J.Math. Combn. Vol.3013), Vertex Graceful Labelng-Some Path Related Graphs P.Selvaraju 1, P.Balaganesan and J.Renuka 3 1 Department of Mathematcs, Vel Tech Engneerng College, Avad, Chenna , Taml Nadu, Inda Department of Mathematcs, School of Engneerng, Saveetha Unversty, Chenna , Taml Nadu, Inda 3 Departments of Mathematcs, Sa Ram College of Engneerng, Chenna , Inda E-mal: pselvarr@gmal.com, balk507@yahoo.co.n Abstract: In ths artcle, we show that an algorthm for VG of a caterpllar and proved that Am j, n) s vertex graceful f m j s monotoncally ncreasng, j n, when n s odd, 1 m 3 and m 1 < m, m j, n) P 3 s vertex graceful f m j s monotoncally ncreasng, j n, when n s odd, 1 m 3, m 1 < m and C n C n+1 s vertex graceful f and only f n 4. Key Words: Vertex graceful graphs, vertex graceful labelng, caterpllar, actna graphs, Smarandachely vertex m-labelng. AMS010): 05C78 1. Introducton A graph G wth p vertces and q edges s sad to be vertex graceful f a labelng f : V G) {1,, 3 p} exsts n such a way that the nduced labelng f + : EG) Z q defned by f + u, v)) = fu) + fv)mod q) s a bsecton. The concept of vertex graceful V G) was ntroduced by Lee, Pan and Tsa n 005. Generally, f replacng q by an nteger m and f S : EG) Z m also s a bjecton, such a labelng s called a Smarandachely vertex m-labelng. Thus a vertex graceful labelng s n fact a Smarandachely vertex q-labelng. All graphs n ths paper are fnte smple graphs wth no loops or multple edges. The symbols V G) and EG) denote the vertex set and edge set of the graph G. The cardnalty of the vertex set s called the order of G. The cardnalty of the edge set s called the sze of G. A graph wth p vertces and q edges s called a p, q) graph.. Man Results Algorthm.1 1. Let v 1, v v n be the vertces of a path n the caterpllar. refer Fgure 1).. Let v j be the vertces, whch are adjacent to v for 1 n and for any j. 3. Draw the caterpllar as a bpartte graph n two partte sets denoted as Left L) whch 1 Receved Aprl 10, 013, Accepted August 15, 013.

2 Vertex Graceful Labelng-Some Path Related Graphs 45 contans v 1, v j, v 3, v 4j, and for any j and Rght R) whch contans v 1j, v, v 3j, v 4, and for any j. refer Fgure ). 4. Let the number of vertces n L be x. 5. Number the vertces n L startng from top down to bottom consecutvely as 1,,, x. 6. Number the vertces n R startng from top down to bottom consecutvely as x + 1),, q. Note that these numbers are the vertex labels. 7. Compute the edge labels by addng them modulo q. 8. The resultng labelng s vertex graceful labelng. v1 1) v 1) v3 3) 18 v4 15) 3 v5 7) v11 v11 v1 v14 v1 v31 v3 v41 v4 v43 51 v5 v53 v54 8) 9) 10) 11) ) 13) 14) 4) 5) 6) 16) 17) 18) 19) 0) v v55 Fgure 1: A caterpllar 1) ) 3) 4) 5) 6) 7) v 1 v 1 v 3 v 41 v 4 v 43 v v 11 v 1 v 13 v 14 v v 31 v 3 v4 v 5 v 51 8) 9) 10) 11) 1) 13) 14) 15) 16) 17) v 18) 53 v 19) 54 v 0) 55 Fgure : A caterpllar as bpartte graph Defnton. The graph Am, n) obtaned by attachng m pendent edges to the vertces of the cycle C n s called Actna graph. Theorem.3 A graph Am j, n), m j s monotoncally ncreasng wth dfference one, j n s vertex graceful, 1 m 3 when n s odd. Proof Let the graph G = Am j, n), m j be monotoncally ncreasng wth dfference one, j n, n be odd wth p = n + m n mn+1 ) m 1 m1+1 ), m 1 = m 1 vertces and q = p edges. Let v 1, v, v 3,, v n be the vertces of the cycle C n. Let v j j = 1,, 3,, n) denote the vertces whch are adjacent to v. By defnton of vertex graceful labelng, the requred

3 46 P.Selvaraju, P.Balaganesan and J.Renuka vertces labelng are 1) m + +1) v = + 1, 1 n, s odd, m + 1) n+1) + n 1 + ) m + +, 1 n, seven. n 1) m + n+1) v j = + 1 m j, 1 j m + 1, s odd; ) m j, 1 m + 1, s even. The correspondng edge set labels are as follows: Let A = {e = v v +1 /1 n 1 e n = v n v 1 }, where m + 1)n + 1) n 1 e = + + m 1) + for 1 n. B = {e j = v v j /1 n}, where n 1) n + 1) e j = m + + 1) m ) + 1 mod q) ) + 1) + + j + 1 mod q) for 1 n and s odd, j = 1,,, m + 1. C = {e j = v v j /1 n}, where n + 1) n 1 e j = m + 1) + + m + 1) + + j mod q) for 1 n and s even, j = 1,,, m + 1. Hence, the nduced edge labels of G are q dstnct ntegers. Therefore, the graph G = Am j, n) s vertex graceful for n s odd, and m 1. Theorem.4 A graph Am j, n) P 3, m j be monotoncally ncreasng, j n s vertex graceful, 1 m 3, n s odd. Proof Let the graph G = Am j, n) P 3, m j be monotoncally ncreasng, j n, m n s odd wth p = n m n+1) m n m 1+1) 1, m 1 < m vertces and q = p 1 edges. Letv 1, v, v 3,, v n be the vertces of the cycle C n. Let v j j = 1,, 3,, n) denote the vertces whch are adjacent to v. Let u 1, u, u 3 be the vertces of the path P 3. By defnton of vertex graceful labelng, the requred vertces labelng are 1 m ; 1 n, s odd; v = m + 1) n+1) + n 1 + ) m ; 1 n, s even. n 1 m + n m j + ; 1 n, s odd, v j = m j + ; 1 n, s even. u = n 1 m + n for = 1, 3 and u = p. The correspondng edge labels are as follows: Let A = {e = v v +1 /1 n 1 e n = v n v 1 }, where m + 1)n + 1) n 1 e = + + m 1) + + 1) + 3 mod q)

4 Vertex Graceful Labelng-Some Path Related Graphs 47 for 1 n. B = {e j = v v j /1 n}, where n 1) n + 1) e j = m + + 1) m + 1 ) + 1) + + j + 3 mod q) for 1 n and s odd, j = 1,,, m + 1. C = {e j = v v j /1 n}, where n + 1) n 1 e j = m + 1) + + m + 1) + + j + mod q) for1 n and s even, j = 1,,, m + 1. D = {e = u u +1 for = 1, }, where n 1 e = m + n mod q) for = 1,. Hence, the nduced edge labels of G are q dstnct ntegers. Therefore, the graph G = Am j, n) P 3 s vertex graceful for n s odd. Defnton.5 A regular lobster s defned by each vertex n a path s adjacent to the path P. Theorem.6 A regular lobster s vertex graceful. Proof Let G be a 1- regular lobster wth 3n vertces and q = 3n 1 edges. Let v 1, v, v 3,, v n be the vertces of a path P n. Let v be the vertces, whch are adjacent to v1 and v 1 adjacent to v for 1 n and n s even.the theorem s proved by two cases. By defnton of Vertex graceful labelng, the requred vertces labelng are Case 1 n s even 3 1 ; 1 n, s odd, v = 3n + ) ; 1 n, s even. 3n + ) 1 / 1 n, s odd v 1 = 3 ) + 3/ 1 n, s even. 3 1) + ; 1 n, s odd, v = 3n + ) 1; 1 n, s even. The correspondng edge labels are as follows: 3n + ) Let A = {e = v v +1 /1 n 1}, where e = + 1 mod q) for 1 n 1, 3n + ) B = {e 1 = v v 1 /1 n}, where e 1 = 1 mod q) for 1 n and s odd, ) 3n + ) C = {e 1 = v v 1 /1 n}, where e 1 = mod q) for 1 n and s even, ) 3n + ) D = {e = v 1 v /1 n}, where e = mod q) for 1 n and s odd, ) 3n + ) E = {e = v 1 v /1 n}, where e = 1 mod q) for 1 n and s even.

5 48 P.Selvaraju, P.Balaganesan and J.Renuka Case n s odd 3 1 ; 1 n, s odd, v = 3n + ) + 1 ; 1 n, s even, 3n+) ; 1 n, s odd, v 1 = 3 ) + 3; 1 n, s even, 3 1) + ; 1 n, s odd, v = 3n + 1) + 1; 1 n, s even. The correspondng edge ) labels are determned by A = {e = v v +1 /1 n 1}, 3n + + 1) where e = mod q) for 1 n 1, B = {e 1 = v v 1 /1 n}, where ) 3n + ) 1 e 1 = mod q) for 1 n and s odd, C = {e 1 = v v 1 /1 n}, where ) 3n + ) + 1 e 1 = mod q) for 1 n and s even, D = {e = v 1 v /1 n}, where ) 3n + ) + 1 e = mod q) for 1 n and s odd, E = {e = v 1v /1 n}, where ) 3n + ) 1 e = mod q) for 1 n and s even. Hence the nduced edge labels of G are q dstnct edges. Therefore, the graph G s vertex graceful. Theorem.7 C n C n+1 s vertex graceful f and only f n 4. Proof Let G = C n C n+1 wth p = n + 1 vertces and q = n + 1 edges. Suppose that the vertces of the cycle C n run consecutvely u 1, u,, u n wth u n joned to u 1 and that the vertces of the cycle C n+1 run consecutvely v 1, v,, v n+1 wth v n+1 joned to v 1. By defnton of vertex graceful labelng a) u 1 = 1, u n =, u = for =, 3,, n + 1)/, u j = n j) + 3 for j = n + 3)/,, n 1. b) v 1 =, v = n 1 and ) v 3s+t = n 4t 6s+7, t = 0, 1,, s = 1,,, n+1 3t)/6 f s = n + 1 3t < 1 6 then no s. ) Wrte α0) = 0, α1) = 4, α) =, β0) = 0, β1) = 3 = β) v n+1 3s t = n 6s αt), t = 0, 1,, s = 0, 1,, n 5 βt). If s = n 5 βt) < then no s value exsts. ) We consder as that v to f); and suppose that n = θ mod3), 0 θ. There are + θ vertces as yet unlabeled. These mddle vertces are labeled accordng to congruence class of modulo 6.

6 Vertex Graceful Labelng-Some Path Related Graphs 49 Congruence class n = 0 mod 6 ) n = 1 mod 6 ) fn + )/) = n +, fn + 4)/) = n + 3, fn + 6)/) = n + 4 fn + 1)/) = n +, fn + 3)/) = n + 3, fn + 5)/) = n + 4, fn + 7)/) = n+5 n = mod 6 ) fn + )/) = n +, fn + 4)/) = n + 3 n = mod 6 ) n = 4 mod 6 ) fn + 1)/) = n+4, fn+3)/) = n+3, fn+5)/) = n+ fn + )/) = n+5, fn+3)/) = n+4, fn+4)/) = n+3, fn+5)/) = n+ n = 4 mod 6 ) fn + 3)/) = n+3, fn+5)/) = n+ To check that f s vertex graceful s very tedous. But we can gve basc dea. The C n cycle has edges wth labels {k+/k = 4, 5,, n 1} {0, 3, 5, 7}. In ths case all the labelng of the edges of the cycle C n+1 run consecutvely v 1 v as follows: 1, n 1, n 3), n 11, n 13, n 15),, n+1 1k, n 1 1k, n 3 1k),, mddle labels,, n+3 1k, n+5 1k, n+7 1k),, n 1, n 19, n 17), n 9, n 7, n 5),. The mddle labels depend on the congruence class modulo and are best summarzed n the followng table. If n s small the terms n brackets alone occur. Congruence class n = 0mod6) 11, 9), 6, 4, 7, 13, 15, 17) n = 1mod6) 13, 11), 6, 4, 7, 13, 15, 17) n = mod6) 11), 6, 4, 7, 9) n = mod6) 13), 7, 4, 6, 9, 11) n = 4mod6) 15, 9), 6, 4, 711, 13) Thus, all these edge labelngs are dstnct. n = 4mod6) 9), 7, 6, 411, 13, 15) References 1 J.A.Gallan, A Dynamc Survey of graph labelng, The Electronc journal of Combnotorcs, ), #DS6. Harary F., Graph Theory, Addson Wesley, Mass Readng, Sn-Mn Lee, Y.C.Pan and Mng-Chen Tsa, On vertex- graceful p,p+1) Graphs, Congressus Numerantum, ), M.A Seoud and A.E.I Abd el Maqsoud, Harmonous graphs, Utltas Mathematca, ), pp P.Balaganesan, P.Selvaraju, J.Renuka,V.Balaj, On vertex graceful labelng, Bulletn of Kerala Mathematcs Assocaton, Vol.9,June 01),