Sum divisor cordial graphs

Transcription

1 Proyeccioes Joural of Mathematics Vol. 35, N o 1, pp , March 016. Uiversidad Católica del Norte Atofagasta - Chile Sum divisor cordial graphs A. Lourdusamy St. Xavier s College, Idia ad F. Patrick St. Xavier s College, Idia Received : December 015. Accepted : March 016 Abstract A sum divisor cordial labelig of a graph G with vertex set V is a bijectio f from V (G) to {1,,, V (G) } such that a edge uv is assiged the label 1 if divides f(u)+f(v) ad 0 otherwise, the the umber of edges labeled with 0 ad the umber of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labelig is called a sum divisor cordial graph. I this paper, we prove that path, comb, star, complete bipartite, K + mk 1, bistar, jewel, crow, flower, gear, subdivisio of the star, K 1,3 K 1, ad square graph of B, are sum divisor cordial graphs. Subjclass : 05C78. Keywords : Sum divisor cordial, divisor cordial.

2 10 A. Lourdusamy ad F. Patrick 1. Itroductio All graphs cosidered here are simple, fiite, coected ad udirected. We follow the basic otatios ad termiologies of graph theory as i []. A labelig of a graph is a map that carries the graph elemets to the set of umbers, usually to the set of o-egative or positive itegers. If the domai is the set of vertices the labelig is called vertex labelig. If the domai is the set of edges, the we speak about edge labelig. If the labels are assiged to both vertices ad edges the the labelig is called total labelig. For a dyamic survey of various graph labelig, we refer to Gallia [1]. Defiitio 1.1. Let G =(V (G),E(G)) be a simple graph ad f : V (G) {1,,, V (G) } be a bijectio. For each edge uv, assig the label 1 if either f(u) f(v) or f(v) f(u) ad the label 0 otherwise. The fuctio f is called a divisor cordial labelig if e f (0) e f (1) 1. A graph which admits a divisor cordial labelig is called a divisor cordial graph. Motivated by the cocept of divisor cordial labelig, we itroduce a ew cocept of divisor cordial labelig called sum divisor cordial labelig. Defiitio 1.. Let G =(V (G),E(G)) be a simple graph ad f : V (G) {1,,, V (G) } be a bijectio. For each edge uv, assig the label 1 if (f(u)+f(v)) ad the label 0 otherwise. The fuctio f is called a sum divisor cordial labelig if e f (0) e f (1) 1. A graph which admits a sum divisor cordial labelig is called a sum divisor cordial graph. Defiitio 1.3. The comb P K 1 is the graph obtaied from a path by attachig a pedat edge to each vertex of the path. Defiitio 1.4. The bistar B, is the graph obtaied by attachig the apex vertices of two copies of K 1, by a edge. Defiitio 1.5. The complete bipartite graph is a simple bipartite graph such that every vertex i oe of the bipartitio subsets is joied to every vertex i the other bipartitio subset. Ay complete bipartite graph that has m vertices i oe of its subsets ad vertices i other is deoted by K,m. Defiitio 1.6. The joi of two graphs G 1 ad G is deoted by G 1 + G ad whose vertex set is V (G 1 + G )=V (G 1 ) V (G ) ad edge set is E(G 1 + G )=E(G 1 ) E(G ) {uv : u V (G 1 ),v V (G )}.

3 Sum divisor cordial graphs 11 Defiitio 1.7. The jewel J is the graph with vertex set V (J )={u, v, x, y, u i : 1 i } ad edge set E(J )={ux, uy, xy, xv, yv, uu i,vu i :1 i }. Defiitio 1.8. The crow C K 1 is the graph obtaied from a cycle by attachig a pedat edge to each vertex of the cycle. Defiitio 1.9. The helm H is the graph obtaied from a wheel by attachig a pedat edge to each rim vertex. The flower Fl is the graph obtaied from a helm by attachig each pedat vertex to the apex of the helm. Defiitio The gear G is the graph obtaied from a wheel by subdividig each of its rim edge. Defiitio K 1,3 K 1, is the graph obtaied from K 1,3 by attachig root of a star K 1, at each pedat vertex of K 1,3. Defiitio 1.1. For a simple coected graph G thesquareofgraphg is deoted by G ad defiedasthegraphwiththesamevertexsetasof G ad two vertices are adjacet i G if they are at a distace 1 or apart i G. Defiitio The subdivisio of star S(K 1, ) is the graph obtaied from K 1, by attachig a pedat edge to each vertex of K 1, except root vertex.. mai results Theorem.1. The path P is sum divisor cordial graph. Proof. Let P be a path with cosecutive vertices v 1,v,,v. The P is of order ad size 1. Defie f : V (P ) {1,,,} as follows: Case 1: is odd i if i 0, 1(mod 4) f(v i )= i +1 ifi (mod 4) for 1 i i 1 if i 3(mod 4)

4 1 A. Lourdusamy ad F. Patrick Case : is eve i if i 1, (mod 4) f(v i )= i +1 ifi 3(mod 4) for 1 i i 1 if i 0(mod 4) I both cases, the iduced edge labels are ( f 1 if (f(vi )+f(v (v i v i+1 )= i+1 )) 0 otherwise We observe that, e f (0) = ( 1 if is odd if is eve ( 1 e f (1) = if is odd if is eve Hece, the path P is sum divisor cordial graph. Example.. A sum divisor cordial labelig of P 6 ad P 7 is show i Figure.1 Theorem.3. The comb is sum divisor cordial graph. Proof. Let G be a comb obtaied from the path v 1,v,,v by joiig a vertex u i to v i for each i =1,,,. The G is of order ad size 1. Defie f : V (G) {1,,, } as follows:

5 Sum divisor cordial graphs 13 f(v i )=i 1; 1 i f(u i )=i;1 i The, the iduced edge labels are f (v i v i+1 )=1;1 i 1 f (v i u i )=0;1 i We observe that, e f (0) = ad e f (1) = 1. Hece, the comb is sum divisor cordial graph. Example.4. A sum divisor cordial labelig of comb is show i Figure. Theorem.5. The star K 1, is sum divisor cordial graph. Proof. Let (V 1,V ) be the bipartitio of K 1, with V 1 = {u} ad V = {u 1,u,,u }. Let E(K 1, )={uu i :1 i }. The K 1, is of order +1 ad size. Defie f : V (K 1, ) {1,,,+1} as follows: f(u) =1; f(u i )=i +1;1 i The, the iduced edge labels are» ¼ f (uu i 1 )=0;1 i

6 14 A. Lourdusamy ad F. Patrick ¹ º f (uu i )=1;1 i We observe that, e f (0) = e f (1) = ( +1 if is odd if is eve ( 1 if is odd if is eve Hece, the star K 1, is sum divisor cordial graph. Example.6. A sum divisor cordial labelig of K 1,5 isshowifigure.3. Theorem.7. The graph K, is sum divisor cordial graph. Proof. Let (V 1,V ) be the bipartitio of K, with V 1 = {u, v} ad V = {v 1,v,,v }. Let E(K, )={uv i,vv i :1 i }. The K, is of order + ad size. Defie f : V (K, ) {1,,,+} as follows: f(u) =1; f(v) =; f(v i )=i +;1 i The, the iduced edge labels are f (uv i 1 )=1;1 i f (uv i )=0;1 i f (vv i 1 )=0;1 i

7 Sum divisor cordial graphs 15 f (vv i )=1;1 i We observe that, e f (0) = e f (1) =. Hece, K, is sum divisor cordial graph. Example.8. A sum divisor cordial labelig of K,5 isshowifigure.4. Theorem.9. The graph K + mk 1 is sum divisor cordial graph. Proof. Let G = K + mk 1. Let V (G) ={u, v, w 1,w,,w m } ad E(G) ={uv, uw i,vw i :1 i m}. The G is of order m + ad size m +1. Defie f : V (G) {1,,,m+} as follows: f(u) =1; f(v) =; f(w i )=i +;1 i m. The, the iduced edge labels are f (uv) =0; f (uw i 1 )=1;1 i m f (uw i )=0;1 i m f (vw i 1 )=0;1 i m f (vw i )=1;1 i m

8 16 A. Lourdusamy ad F. Patrick We observe that, e f (0) = m +1ade f (1) = m. Hece, K + mk 1 is sum divisor cordial graph. Example.10. A sum divisor cordial labelig of K +5K 1 is show i Figure.5. Theorem.11. The bistar B, is sum divisor cordial graph. Proof. Let G = B,. Let V (G) = {u, v, u i,v i : 1 i } ad E(G) ={uv, vv i,uu i :1 i }. The G is of order + ad size +1. Defie f : V (G) {1,,, +} as follows: f(u) =1; f(v) =; f(u i 1 )=4i 1; 1 i f(u i )=4i +;1 i f(v i 1 )=4i;1 i f(v i )=4i +1;1 i The, the iduced edge labels are f (uv) =0; f (uu i 1 )=1;1 i f (uu i )=0;1 i f (vv i 1 )=1;1 i We observe that, f (vv i )=0;1 i ( if is odd e f (0) = +1 if is eve

9 Sum divisor cordial graphs 17 Thus, e f (0) e f (1) 1. ( +1 if is odd e f (1) = if is eve Hece, the bistar B, is sum divisor cordial graph. Example.1. A sum divisor cordial labelig of B 5,5 isshowifigure.6. Theorem.13. The flower Fl is sum divisor cordial graph. Proof. Let G = Fl. Let V (G) = {v, v i,u i : 1 i } ad E(G) ={vv i,v i u i,vu i :1 i ; v v 1 ; v i v i+1 :1 i 1}. The G is of order +1adsize4. Defie f : V (G) {1,,, +1} as follows: f(v) =1; f(v i )=i;1 i f(u i )=i +1;1 i The, the iduced edge labels are f (vv i )=0;1 i f (vu i )=1;1 i f (v i u i )=0;1 i f (v i v i+1 )=1;1 i 1 f (v v 1 )=1;

10 18 A. Lourdusamy ad F. Patrick We observe that, e f (0) = e f (1) =. Thus, e f (0) e f (1) 1 Hece, Fl is sum divisor cordial graph. Example.14. A sum divisor cordial labelig of Fl 4 is show i Figure.7. Theorem.15. The jewel J is sum divisor cordial graph. Proof. Let G = J. Let V (G) = {u, v, x, y, u i : 1 i } ad E(G) ={ux,uy,xy,xv,yv,uu i,vu i :1 i }. TheG is of order +4 ad size +5. Defie f : V (G) {1,,,+4} as follows: f(u) =1; f(v) =; f(x) =3; f(y) =4; f(u i )=i +4;1 i. The, the iduced edge labels are f (ux) =1; f (uy) =0; f (xy) =0; f (vx) =0; f (vy) =1; f (uu i 1 )=1;1 i f (uu i )=0;1 i f (vu i 1 )=0;1 i

11 Sum divisor cordial graphs 19 f (vu i )=1;1 i We observe that, e f (0) = +3ade f (1) = +. Hece, J is sum divisor cordial graph. Example.16. A sum divisor cordial labelig of J 4 is show i Figure.8. Theorem.17. The crow C K 1 is sum divisor cordial graph. Proof. Let G = C K 1. Let V (G) = {u i,v i : 1 i } ad E(G) ={u i u i+1 :1 i 1; u u 1,u i v i :1 i }. TheG is of order ad size. Defie f : V (G) {1,,, } as follows: f(u i )=i;1 i f(v i )=i 1; 1 i The, the iduced edge labels are f (u i u i+1 )=1;1 i 1 f (u u 1 )=1; f (u i v i )=0;1 i We observe that, e f (0) = e f (1) =. Hece, C K 1 is sum divisor cordial graph.

12 130 A. Lourdusamy ad F. Patrick Example.18. A sum divisor cordial labelig of C 4 K 1 is show i Figure.9. Theorem.19. The gear G is sum divisor cordial graph. Proof. Let G = G. Let V (G) = {v, u i,v i : 1 i } ad E(G) ={vv i,v i u i :1 i ; u i v i+1 :1 i 1; u v 1 }. The G is of order +1 ad size 3. Defie f : V (G) {1,,, +1} as follows: f(v) =1; f(v i 1 )=4i 1; 1 i f(v i )=4i;1 i f(u i 1 )=4i ; 1 i f(u i )=4i +1;1 i The, the iduced edge labels are f (vv i 1 )=1;1 i f (vv i )=0;1 i f (v i u i )=0;1 i f (u i v i+1 )=1;1 i 1 ( f 0 if is odd (u v 1 )= 1 if is eve

13 Sum divisor cordial graphs 131 l m j k We observe that, e f (0) = 3 ad e f (1) = 3 Hece, G is sum divisor cordial graph. Example.0. A sum divisor cordial labelig of G 8 isshowifigure.10. Theorem.1. The graph K 1,3 K 1, is sum divisor cordial graph. Proof. Let G = K 1,3 K 1,. Let V (G) ={x, u, v, w, u i,v i,w i :1 i } ad E(G) ={xu, xv, xw, uu i,vv i,ww i :1 i }. TheG is of order 3 +4adsize3 +3. Defie f : V (G) {1,,, 3 +4} as follows: f(u) =1; f(v) =; f(w) =4; f(x) =3; f(u i )=3i +;1 i f(v i )=3i +3;1 i f(w i )=3i +4;1 i The, the iduced edge labels are f (xu) =1;

14 13 A. Lourdusamy ad F. Patrick We observe that, f (xv) =0; f (xw) =0; f (uu i 1 )=1;1 i f (uu i )=0;1 i f (vv i 1 )=1;1 i f (vv i )=0;1 i f (ww i 1 )=0;1 i f (ww i )=1;1 i e f (0) = ( 3+3 if is odd 3+4 if is eve ( 3+3 e f (1) = if is odd 3+ if is eve Hece, K 1,3 K 1, is sum divisor cordial graph. Example.. A sum divisor cordial labelig of K 1,3 K 1,5 is show i Figure.11. Theorem.3. The graph B, is sum divisor cordial graph.

15 Sum divisor cordial graphs 133 Proof. Let G = B,. Let V (G) = {u, v, u i,v i : 1 i } ad E(G) ={uv, vv i,uu i,u i v, v i u :1 i }. TheG is of order +ad size Defie f : V (G) {1,,, +} as follows: f(u) =1; f(v) =; f(u i 1 )=4i 1; 1 i f(u i )=4i +;1 i f(v i 1 )=4i;1 i The, the iduced edge labels are f (uv) =0; f(v i )=4i +1;1 i f (uu i 1 )=1;1 i f (uu i )=0;1 i f (vv i 1 )=1;1 i f (vv i )=0;1 i f (vu i 1 )=0;1 i f (vu i )=1;1 i f (uv i 1 )=0;1 i f (uv i )=1;1 i We observe that, e f (1) = ad e f (0) = +1. Hece, the graph B, is sum divisor cordial graph. Example.4. A sum divisor cordial labelig of B 4,4 isshowifigure.1.

16 134 A. Lourdusamy ad F. Patrick Theorem.5. The graph S(K 1, ) is sum divisor cordial graph. Proof. Let G = S(K 1, ). Let V (G) ={v, v i,u i :1 i } ad E(G) ={vv i,v i u i :1 i }. The G is of order +1 ad size. Defie f : V (G) {1,,, +1} as follows: f(v) =1; f(v i )=i +1;1 i f(u i )=i;1 i The, the iduced edge labels are f (vv i )=1;1 i f (v i u i )=0;1 i We observe that, e f (0) = e f (1) =. Hece, S(K 1, ) is sum divisor cordial graph. Example.6. A sum divisor cordial labelig of S(K 1,5 ) is show i Figure coclusio All the graphs are ot sum divisor cordial graphs. It is very iterestig ad challegig as well to ivestigate sum divisor cordial labelig for the

17 Sum divisor cordial graphs 135 graph or graph families which admit sum divisor cordial labelig. Here we have proved path, comb, star, complete bipartite, K + mk 1,bistar,jewel, crow, flower, gear, subdivisio of the star, K 1,3 K 1, ad square graph of B, are sum divisor cordial graphs. I the subsequet paper, we will prove that total graph of the path, square graph of the path, shadow graph of the path ad alterative triagular sake are sum divisor cordial graphs. Also, we will prove book, oe poit uio of cycles, triagular ladder related graphs are sum divisor cordial graphs. Refereces [1] J. A. Gallia, A Dyamic Survey of Graph Labelig, The Electroic J. Combi., 17 (015) #DS6. [] F. Harary, Graph Theory, Addiso-wesley, Readig, Mass (197). [3] P. Lawrece Rozario Raj ad R. Lawece Joseph Maohara, Some ResultoDivisorCordialLabeligofGraphs,It.J.IocativeSci.,1 (10), pp. 6-31, (014). [4] P. Maya ad T. Nicholas, Some New Families of Divisor Cordial Graph, Aals Pure Appl. Math., 5 (), pp , (014). [5] A. Nellai Muruga ad G. Devakiruba, Cycle Related Divisor Cordial Graphs, It. J. Math. Treds ad Tech., 1 (1), pp , (014). [6] A. Nellai Muruga, G. Devakiruba ad S. Navaaeethakrisha, Star Attached Divisor Cordial Graphs, It. J. Io. Sci. Egieerig ad Tech., 1 (5), pp , (014). [7] S. K. Vaidya ad N. H. Shah, Some Star ad Bistar Related Cordial Graphs, Aals Pure Appl. Math., 3 (1), pp , (013). [8] S. K. Vaidya ad N. H. Shah, Further Results o Divisor Cordial Labelig, Aals Pure Appl. Math., 4 (), pp , (013). [9] R. Varatharaja, S. Navaaeethakrisha ad K. Nagaraja, Divisor Cordial Graphs, It. J. Math. Combi., 4, pp. 15-5, (011).

18 136 A. Lourdusamy ad F. Patrick A. Lourdusamy Departmet of Mathematics, St. Xavier s College, Palayamkottai-6700, Tamiladu, Idia lourdusamy15@gmail.com ad F. Patrick Departmet of Mathematics, St. Xavier s College, Palayamkottai-6700, Tamiladu, Idia patrick881990@gmail.com