Modulo Magic Labeling in Digraphs

Transcription

1 Gen. Math. Notes, Vol. 7, No., August, 03, pp. 5- ISSN 9-784; Copyrght ICSRS Publcaton, 03 Avalable free onlne at Modulo Magc Labelng n Dgraphs L. Shobana and J. Baskar Babujee, Department of Mathematcs, Anna Unversty Chenna , Inda E-mal: shobana_0@yahoo.com E-mal: baskarbabujee@yahoo.com (Receved: 5-6- / Accepted: 0-6-3) Abstract In ths paper we ntroduce and study a new type of labelng called Modulo Magc Labelng. Keywords: Dgraphs, Congruence relaton, Cycles, Path, Labelng. Introducton A drected graph G = (V,E) conssts of a fnte set V of ponts (vertex) and a collecton of ordered par of dstnct ponts called lnes (edges). Any such par ( u, v) s called an arc or drected lne (edge) and wll usually be denoted by uv. The edge uv goes from u to v and ncdent wth u and v. An edge connects a vertex to tself s called a loop. Let us consder the set H to be H = { 0,,,3,..., n }. We consder the drected graph whose vertces are the elements of H and such that there exsts exactly one drected edge from a to b f and only f t satsfes the congruence relaton. A cycle of length s called a fxed pont. Denote by d(a) the number of drected edges comng to a and a) the number of drected edges leavng the vertex a H = { 0,,,3,..., n }.. Of course

2 6 L. Shobana et al. d(a) 0 and od (a) =. For an solated fxed pont, the n-degree and the outdegree are both equal to. In the followng sectons we show some connecton between congruence relaton and graph labelng motvated by results of M.Krzek and L.Somer gven n [8] and n [9] as well as by the results of S.Bryant[5], G.Chasse[6], T.D.Rogers[], L.Szalay[] and Joanna Skowronek-Kazow[7]. The followng defntons are taken from [0] and [3]. Defnton.: Let G be a graph wth p vertces and q edges. Assume that the vertces and edges of G are labeled by,, 3... p + q such that each label s used exactly once. We defne the valence e) of an edge e to the sum of the label of e plus the two labels of the vertces ncdent wth e. Defnton.: If a labelng of G s possble such that the valence e) for each e s constant, we call the graph G edge-magc. And f t s possble that all the valences of the edge are dstnct, we call G edge-antmagc. Defnton.3: Let the edges of a graph G be labeled by,,..., q, and we call the sum of all the labels of the edges ncdent wth a vertex P the valence P) of P. If t s possble that all of the valences of the vertces n G are dstnct, the graph G s called antmagc. Defnton.4: A graph G(p, q) s sad to be (, ) vertex-magc wth the common vertex count k f there exsts a bjecton f : V(G) E(G) {,..., p + q} such that for each u V(G), f(u) + e f(e) = k for all e = (u, v) E(G) wth v V (G). It s sad to be (,) vertex- antmagc f f(u) + e f(e) are dstnct for all e = (u, v) E(G) wth v V(G). (, ) vertex magc labelng can also be sad as vertex magc total labelng. Modulo Magc Labelng In ths secton we ntroduce the concept of modulo magc labelng for some congruence relatons wth constant valence n-. The out-degree of a vertex v s the number of edges gong away from v n a drected graph and s denoted by v ). The valence of v ) s the sum of the out degree labels of edges from v plus the label of v. Defnton.: Let G be a drected graph wth = { 0,,,3,..., n } bjectve functon : V E { 0,,,3,..., n } V and a f s sad to be a modulo magc labelng wth common count k, f f ( u) + f ( uv) = k uv E or val ( od (u) ) s a constant k.

3 Modulo Magc Labelng n Dgraphs 7 Theorem.: If G(V,E) be a drected graph wth = { 0,,,..., n } V and edge set ab E f and only f arb when R : a + b mod n admts modulo magc labelng. Proof: Let G = ( V, E ) be a graph wth vertex set = { 0,,,3,..., n } V and the edge set s defned by the relaton a + b mod n. Consder a bjectve functon f : V E {0,,,..., n, n,...,n } defned by f ( v ) = ; 0 n and the edge set s gven by { vv+ ; 0 n } { v 0 v } E. The edge labelng s defned by = n f ( v v+ ) = n ; 0 n and f ( v n v0 ) = n. The valence of out-degree of the vertex v s denoted by val ( od ( v )). For every, v )) = f ( v ) + f ( v v + ); 0 n = + n = n vn )) = f ( vn ) + f ( vn v0 ) = n + n = n Hence the drected cycle graph obtaned by the relaton modulo magc labelng wth constant value n. a + bmod n admts Theorem.3: If G(V,E) be a dsconnected drected cycle graph wth { 0,,,..., } V = n and edge set ab E f and only f arb when R : a + b mod n admts modulo magc labelng. Proof: Let G = ( V, E) be a graph wth vertex set = { 0,,,3,..., n } V and the edge set s defned by the relaton a + b mod n. Consder a functon f : V E {0,,,..., n, n,...,n } where f ( v ) = ; 0 n. We now proceed wth the followng two cases. Case : When n 0 mod Let G = ( V, E) be a dsconnected graph. The two drected cycle graphs obtaned from the relaton s symmetrc. The edge set s defned by { vn v 0 } { vn } E = E where E = { v v + ; 0 ( n 4) / } and E v

4 8 L. Shobana et al. E = { v v + ; ( n )/ }. The edge labelng are gven by f ( v v ) =n--; 0 ( n 4)/ + f( v v + )=n-; ( n )/ f ( vn v0) = n + f vn v = n ( ) v )) = f ( v ) + f ( vv+ ); 0 ( n 4)/ = + n = n v+ )) = f ( v ) + f ( v v+ ); ( n )/ = + n = n vn )) = f ( vn ) + f ( vn v0 ) = n + n + = n vn )) = f ( vn ) + f ( vn v) = n + n = n Case : When n mod Let G ( V, E) = be a connected cycle graph. The edge set s defned by { vn v 0 } { vn } E = { v v v. The edge labelng s gven by + ; 0 n 3} f ( v v ) = n ;0 3 + n f ( vn v0) = n + f vn v = n ( ) v )) = f ( v ) + f ( vv+ ); 0 n 3 = + n = n vn )) = f ( vn ) + f ( vn v0 ) = n + n +

5 Modulo Magc Labelng n Dgraphs 9 = n vn )) = f ( vn ) + f ( vn v) = n + n = n Hence the drected cycle graph obtaned by the relaton magc labelng wth constant valence n a + bmodn admts modulo Example.4: The followng fgure shows a drected graph for the congruence relaton a 7 + b mod0 whch has modulo magc labelng wth constant valence 9. 7 Fg : Modulo magc labelng for a + b mod 0 Observaton.5: Let ( V, E) G = be a graph wth vertex set V ={ 0,,,.,n- } and the edge set s defned by the relaton a m + k b mod n where m, k and n are postve ntegers. Let f : V E {0,,,..., n, n,...,n } defned by f ( v ) = ; 0 n. Let us assume that a m + k b mod n and a m + k c mod n. a m + k b = nx and a m + k c = ny where x and y are postve ntegers. a m + k = b + nx and a m + k = c + ny b + nx = c + ny b c = n( y x) b c mod n. Therefore the out-degree for all the vertces n the drected graph s one. Now the edge labels are defned n such a way that the out degree from v ; 0 n wll

6 0 L. Shobana et al. receve the edge labels n the descendng from n- to n respectvely. Therefore when we sum the vertex wth ther correspondng edge labels we get a constant value n- whch leads to the modulo magc labelng. 3 Modulo Bmagc Labelng In ths secton we ntroduce the concept of modulo bmagc labelng for some congruence relatons. The n-degree of a vertex v s the number of edges comng towards v n a drected graph and s denoted by d(v ). The valence of d(v ) s the sum of the n-degree labels of the vertces v j for some j. Defnton 3.: Let V = { 0,,,3,..., n }. A drected graph G(V,E) s sad to have modulo bmagc labelng wth common count 0 or k, f there exst a bjecton f : V V such that f ( v) = 0 or k uv E. uv Example 3.: The followng fgure shows a drected graph for the congruence relaton a b mod3 whch admts modulo bmagc labelng. Fg : Modulo bmagc labelng for a b(mod 3) Theorem 3.3: If G (V, E) be a drected graph wth V { 0,,,..., n } = and edge set ab E f and only f arb when R : a b mod n admts modulo bmagc labelng only when n s prme. Proof: Let V { 0,,..., n } = be the vertces of the dgraph where n s any prme greater than two. The edges of the graph s defned by the relaton a b(mod n). The graph obtaned wth ths relaton s a dsconnected graph whch s ether a path or a cycle, for whch the n- degree sum of every vertex s ether 0 or n. The

7 Modulo Magc Labelng n Dgraphs pendant vertces have ndegree zero and all the nternal vertces have n degree and 0 s always an solated fxed pont. Let us assume that a b mod n and a c mod n. a b = nk and a c = nm where k and m are postve ntegers. a = b + nk and a = c + nm b + nk = c + nm b c = n( m k) b c mod n whch shows that the out-degree s always one. Consder an arbtrary vertex x whose n degree s. Let u and v be the vertces such that ux, vx E and f ( u) + f ( v) = n where u, v V. Let f ( u) = a, f ( v) = a and f ( x) = b say, then a b mod n () a b mod n. () We need to prove that a + a =. From () and () we nfer that n a a mod n (3) a a 0 mod n a a = kn ( a )( a a a + ) = kn where k s any nteger. From (3) t s clear that a + a = n where a a = k. The vertex labeled wth 0 s an solated fxed pont whose ndegree vertex sum s always 0.Therefore the graph obtaned by the relaton R admts (, 0) modulo bmagc labelng wth two constants 0 and n. Observaton 3.4: A component of the teraton dgraph s a subdgraph whch s a maxmal connected subgraph of the symmetrzaton of ths dgraph. The dgraph G(n) s called symmetrc f ts set of components can be splt nto two sets n such a way that there exsts a bjecton between these two sets such that the correspondng dgraphs are somorphc[8]. Consder the dgraph obtaned by the congruence relaton a + b mod n for n 0. when n s even, the dgraph obtaned by the relaton s symmetrc except when n=8. When n s odd, we get a connected dgraph except for n=5 and 9.

8 L. Shobana et al. 4 Concluson The dgraph constructed from some congruence relatons satsfes the modulo magc labelng and modulo bmagc labelng. In ths smlar way, one can construct and analyze the propertes of the dgraph obtaned from the congruence relatons and also ft the sutable labelng to t. References [] J.B. Babujee, Bmagc labelng n path graphs, The Mathematcs Educaton, 38() (004), -6. [] J.B. Babujee and N.P. Rao, Labelngs n dgraph, Journal of the Mathematcs Educaton, 39() (005), [3] J.B. Babujee and L. Shobana, Dgraphs arsng from number theory, Indan Journal of Mathematcs and Mathematcal Scences, 5() (Dec) (009), [4] J.B. Babujee and L. Shobana, Graph from relatons and ts labelng, European Journal of Scentfc Research, 64(3) (0), [5] S. Bryant, Groups, graphs and Fermat s last theorem, Amer. Math. Monthly, 74(967), [6] G. Chasse, Combnatoral cycles of a polynomal map over a commutatve feld, Dscrete Math., 6(986), -6. [7] J. Skowronek-Kazow, Some dgraphs arsng from number theory and remarks on the zero-dvsor graph of the rng z n, Informaton Processng Letters, 08 (3)(008), [8] M. Krzek and L. Somer, On a connecton of number theory wth graph theory, Czechoslovak Math. J., 54(9) (004), [9] M. Krzek and L. Somer, Sophe German lttle suns, Math. Slovaca, 54(004), [0] G. Rngel, Another tree conjecture, Utltas Matematca, 53(998), [] T.D. Rogers, The graph of the square mappng on the prme felds, Dscrete Math., 48(996), [] L. Szalay, A dscrete teraton n number theory, BDTF Tud. Kozl., 8(99), 7-9 (n Hungaran). [3] V. Yegnanarayanan, On magc graphs, Utltas Matematca, 59(00), [4] W.D. Walls, Magc Graphs, Brkhauser, (000).