Fans are cycle-antimagic

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 68(1 (017, Pages Fas are cycle-atimagic Ali Ovais Muhammad Awais Umar Abdus Salam School of Mathematical Scieces GC Uiversity, Lahore Pakista Marti Bača Adrea Semaičová-Feňovčíková Departmet of Applied Mathematics ad Iformatics Techical Uiversity, Košice Slovakia Abstract A simple graph G =(V,E admits a H-coverig if every edge i E belogs at least to oe subgraph of G isomorphic to a give graph H. The the graph G admittig a H-coverig is (a, d-h-atimagic if there exists a bijectio f : V E {1,,..., V + E } such that, for all subgraphs H of G isomorphic to H, the H -weights, wt f (H = v V (H f(v+ e E(H f(e, form a arithmetic progressio with the iitial term a ad the commo differece d. Such a labelig is called super if the smallest possible labels appear o the vertices. This paper is devoted to studyig the existece of super (a, d-hatimagic labeligs for fas whe subgraphs H are cycles. 1 Itroductio We cosider fiite ad simple graphs. Let the vertex ad edge sets of a graph G be deoted by V = V (G ade = E(G, respectively. A edge-coverig of G is a family of subgraphs H 1,H,...,H t such that each edge of E belogs to at least oe of the subgraphs H i, i =1,,...,t.TheitissaidthatG admits a (H 1,H,...,H t - (edge coverig. If every subgraph H i is isomorphic to a give graph H, the the Also at Abdus Salam School of Mathematical Scieces, GC Uiversity, Lahore, Pakista. Correspodig author. Also at Abdus Salam School of Mathematical Scieces, GC Uiversity, Lahore, Pakista.

2 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, graph G admits a H-coverig. Note that i this case all subgraphs of G isomorphic to H must be i the H-coverig. A bijective fuctio f : V E {1,,..., V + E } is a (a, d-h-atimagic labelig of a graph G admittig a H-coverig wheever, for all subgraphs H isomorphic to H, theh -weights wt f (H = f(v+ f(e v V (H e E(H form a arithmetic progressio a, a + d, a +d,...,a+(t 1d, wherea>0ad d 0 are two itegers, ad t is the umber of all subgraphs of G isomorphic to H. Such a labelig is called super if the smallest possible labels appear o the vertices. A graph that admits a (super (a, d-h-atimagic labelig is called (super (a, d-h-atimagic. Ford = 0 it is called H-magic ad H-supermagic, respectively. The otio of H-supermagic graphs was itroduced by Gutiérrez ad Lladó [8]as a extesio of the edge-magic ad super edge-magic labeligs itroduced by Kotzig ad Rosa [11] ad Eomoto, Lladó, Nakamigawa ad Rigel [7], respectively. They proved that some classes of coected graphs are H-supermagic, such as the stars K 1, ad the complete bipartite graphs K,m are K 1,h -supermagic for some h. They also proved that the path P ad the cycle C are P h -supermagic for some h. More precisely they proved that the cycle C is P h -supermagic for ay h 1such that gcd(, h(h 1 = 1. Lladó ad Moragas [1] studied the cycle-(supermagic behavior of several classes of coected graphs. They proved that wheels, widmills, books ad prisms are C h -magic for some h. Maryati, Salma, Baskoro, Rya ad Miller [16] ad also Salma, Ngurah ad Izzati [18] proved that certai families of trees are path-supermagic. Ngurah, Salma ad Susilowati [17] proved that chais, wheels, triagles, ladders ad grids are cycle-supermagic. Maryati, Salma ad Baskoro [15] ivestigated the G-supermagicess of a disjoit uio of c copies of agraphg ad showed that the disjoit uio of ay paths is cp h -supermagic for some c ad h. The (a, d-h-atimagic labelig was itroduced by Iayah, Salma ad Simajutak [9]. I [10] the authors ivestigate the super (a, d-h-atimagic labeligs for some families of coected graphs H. I [19] was proved that wheels W, 3, are super (a, d-c k -atimagic for every k =3, 4,..., 1,+1add =0, 1,. The (super (a, d-h-atimagic labelig is related to a super d-atimagic labelig of type (1, 1, 0 of a plae graph that is the geeralizatio of a face-magic labelig itroduced by Lih [13]. Further iformatio o super d-atimagic labeligs ca be foud i [, 5]. For H = K,(super(a, d-h-atimagic labeligs are also called (super (a, d- edge-atimagic total labeligs ad have bee itroduced i [0]. More results o (a, d-edge-atimagic total labeligs, ca be foud i [4, 14]. The vertex versio of these labeligs for geeralized pyramid graphs is give i [1]. The existece of super (a, d-h-atimagic labeligs for discoected graphs is studied i [6] ad there is proved that if a graph G admits a (super (a, d-hatimagic labelig, where d = E(H V (H, the the disjoit uio of m copies

3 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, of the graph G, deoted by mg, admits a (super (b, d-h-atimagic labelig as well. I [3] is show that the disjoit uio of multiple copies of a (super (a, 1-treeatimagic graph is also a (super (b, 1-tree-atimagic. A atural questio is whether the similar result holds also for aother differeces ad aother H-atimagic graphs. A fa F,, is a graph obtaied by joiig all vertices of the path P to a further vertex, called the cetre. The vertices o the path we will call the path vertices. The edges adjacet to the cetral vertex we will call the spokes ad the remaiig edges we will call the path edges. Thus F cotais + 1 vertices, say, v 1,v,...,v +1,ad 1edges,say,v +1 v i,1 i, adv i v i+1,1 i 1. I this paper we ivestigate the existece of super (a, d-h-atimagic labeligs for fas whe subgraphs H are cycles. Super (a, d-cycle-atimagic labelig of fa Let C k be a cycle o k vertices. Every cycle C k i F is of the form C j k = v jv j+1 v j+... v j+k v +1 v j,wherej =1,,..., k +. It is easy to see that each edge of F belogs to at least oe cycle C j k if k =3, 4,..., +. For the C k -weight of the cycle C j k, j =1,,..., k +, uder a total labelig f we get wt f (C j k = f(v+ f(e v V (C j k e E(C j k k 3 ( = f(v j+s +f(v j+s v j+s+1 + ( f(v j+k +f(v j+k v +1 + f(v +1 +f(v j v +1. (1.1 Differeces d =1, 3 The ext theorem shows that F admits super cycle-atimagic labeligs for differeces d =1add =3. Theorem 1. Let 3 be a positive iteger ad 3 k +. The the fa F admits a super (a, d-c k -atimagic labelig for d =1, 3. Proof. Let us cosider the total labeligs f 1 ad f of F defied i the followig way f 1 (v i =f (v i =i, for i =1,,...,+1 f 1 (v i v +1 = + i, for i =1,,..., f (v i v +1 = +1+i, for i =1,,..., f 1 (v i v i+1 =f (v i v i+1 =3 +1 i, for i =1,,..., 1.

4 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, It is easy to see that f 1 ad f are super labeligs as the vertices of F are labeled by the labels 1,,...,+1. Uder both labeligs the spokes attai the labels +,+3,..., +1ad the path edges are labeled by the umbers +, +3,...,3. The sum of the path vertex label ad the correspodig icidet path edge label is a costat. More precisely, for every i =1,,..., 1adform =1, wehave f m (v i +f m (v i v i+1 =i +(3 +1 i =3 +1. ( Uder the labelig f 1 the sum of the path vertex label ad the icidet spoke label is a costat, that is, for every i =1,,..., f 1 (v i +f 1 (v i v +1 =i +( + i = +. (3 O the other side uder the labelig f the sums of the path vertex label ad correspodig spoke label form a arithmetic sequece with differece, that is, for every i =1,,..., f (v i +f (v i v +1 =i +( +1+i = +1+i. (4 Accordig to (1, ( ad (3 we obtai wt f1 (C j k =(k (3 +1+( ++( +1+( + j =(k ( j ad with respect to (1, ( ad (4 we obtai wt f (C j k =(k (3 +1+( +1+(j + k + ( +1+( +1+j =(k 1(3 +3+3j. Thus uder the labelig f 1 the set of all the C k -weights cosists of cosecutive itegers ad uder the labelig f the C k -weights form the arithmetic sequece with the differece 3. This cocludes the proof.. Differeces depedig o the legth of cycle The followig theorem proves the existece of super cycle-atimagic labeligs for differeces k 5, k 1ad3k 1. Theorem. Let 3 be a positive iteger ad 3 k +. The the fa F admits a super (a, d-c k -atimagic labelig for d =k 5, k 1, 3k 1. Proof. Let us cosider the total labeligs f 3,f 4 ad f 5 of F defied i the followig

5 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, way f m (v i =i, for i =1,,...,+1adm =3, 4, i, for i =1,,..., ad m =3 f m (v i v +1 = + i, for i =1,,..., ad m =4 +i, for i =1,,..., ad m =5 { f m (v i v i+1 = +1+i, for i =1,,..., 1adm =3, 4 +1+i, for i =1,,..., 1adm =5. It is easy to see that f m is a super labelig for every m =3, 4, 5. Uder the labeligs f 3 ad f 4 the path edges are labeled with the umbers +,+3,..., ad uder the labelig f 5 they attai the umbers +3,+5,...,3 1. The labeligs f 3 ad f 4 assig to spokes the labels +1, +,...,3 ad the labelig f 5 assigs labels +,+4,...,3. For every i =1,,..., 1wehave f m (v i +f m (v i v i+1 =i +( +1+i = +1+i, if m =3, 4 (5 f 5 (v i +f 5 (v i v i+1 =i +( +1+i = +1+3i. (6 For every i =1,,..., we get f 3 (v i +f 3 (v i v +1 =i +(3 +1 i =3 +1, (7 f 4 (v i +f 4 (v i v +1 =i +( + i = +i, (8 f 5 (v i +f 5 (v i v +1 =i +( +i = +3i. (9 For C k -weights from (1, (5 ad (7 it follows k 3 wt f3 (C j k = ( +1+(j + s +(3 +1+( +1+(3 +1 j =(k ( + k j(k 5, by (1, (5 ad (8 we obtai k 3 wt f4 (C j k = ( +1+(j + s +( +(j + k + ( +1+( + j =(k ( + k+5 +1+j(k 1, ad by (1, (6 ad (9 we get k 3 wt f5 (C j k = ( +1+3(j + s +( +3(j + k + ( +1+( +j 3(k 3(k =(k +1( j(3k 1. Thus uder the labeligs f m, m =3, 4, 5, the C k -weights form the arithmetic sequece with the differeces k 5, k 1ad3k 1, respectively.

6 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, The existece of super cycle-atimagic labeligs of a fa for differeces 3k 9, k 7adk + 1 follows from the ext theorem. Note that for some of these values of differece d is egative, which oly meas that the cycle-weights form decreasig sequece, or alteratively the differece i the correspodig icreasig arithmetic sequece is d. Notethatifd = 0 the the cycle-weights are the same. Theorem 3. Let 3 be a positive iteger ad 3 k +. The the fa F is super (a, d-c k -atimagic for d =3k 9,k 7,k+1. Proof. Defie the total labelig f m, m =6, 7, 8, of F as follows i, for i =1,,...,+1adm =6 f m (v i = +1 i, for i =1,,..., ad m =7, 8 +1, for i = +1adm =7, 8 { 3 + i, for i =1,,..., ad m =6, 7 f m (v i v +1 = +i, for i =1,,..., ad m =8 f m (v i v i+1 = +1+i, for i =1,,..., 1adm =6, 7, 8. Sice the labeligs f m, m =6, 7, 8, assig the smallest possible labels to the vertices of F, they are super. For each labelig f m, m =6, 7, 8, the path edges attai the labels +3,+5,...,3 1 ad the spokes are labeled by the labels +,+4,...,3. For every i =1,,..., 1weget f 6 (v i +f 6 (v i v i+1 =i +( +1+i = +1+3i, (10 f m (v i +f m (v i v i+1 =( +1 i+( +1+i = ++i, if m =7, 8 (11 For every i =1,,..., we have f 6 (v i +f 6 (v i v +1 =i +(3 + i =3 + i, (1 f 7 (v i +f 7 (v i v +1 =( +1 i+(3 + i =4 +3 3i, (13 f 8 (v i +f 8 (v i v +1 =( +1 i+( +i = +1+i. (14 Accordig to (1, (10 ad (1 k 3 wt f6 (C j k = ( +1+3(j + s +(3 + (j + k + ( +1 +(3 + j =(k +5+ 3(k 3(k +5+j(3k 9,

7 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, with respect to (1, (11 ad (13 k 3 wt f7 (C j k = ( ++(j + s +(4 +3 3(j + k + ( +1 +(3 + j =( 1(k ++ + j(k 7. ad from (1, (11 ad (14 it follows k 3 wt f8 (C j k = (k 3(k +10 ( ++(j + s +( +1+(j + k + ( +1 (k 3(k +( +j =( +3k + 4+j(k +1. Oe ca see that uder the labeligs f m, m =6, 7, 8, the C k -weights costitute the arithmetic sequeces with the differeces 3k 9, k 7adk + 1, respectively. 3 C 3 -atimagicess of fas I [13] Lih proved that F is C 3 -supermagic for every except whe (mod4. Ngurah, Salma ad Susilowati [17] completed this result ad they proved that for ay iteger thefaf is C 3 -supermagic. Immediately from Theorems 1 through 3 we obtai that if F satisfies the ecessary coditio for coverig by C 3, the there exist super (a, d-c k -atimagic labeligs of F for every d {0, 1, 3, 4, 5, 8}. Moreover, i the ext theorem we are able to prove also that differeces d =add = 6 are feasible. Theorem 4. The fa F, 4, issuper(a, d-c 3 -atimagic for d =0, 1,, 3, 4, 5, 6, 8. Proof. The existece of such labeligs for d =0, 1, 3, 4, 5, 8 immediately follows from Theorems 1 through 3. For d =, 6 let us cosider the followig. Costruct the total labeligs g m, m =1,, of F such that i+1, for 1 i, i 1 (mod ad m =1 + i, for i, i 0 (mod ad m =1 g m (v i = +1, for i = +1adm =1 i, for i =1,,...,+1adm = + i, for i =1,,..., ad m =1 g m (v i v +1 = +1+i, for 1 i, i 1 (mod ad m = + + i, for i, i 0 (mod ad m = { +1 i, for i =1,,..., 1adm =1 g m (v i v i+1 = +1+i, for i =1,,..., 1adm =.

8 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, The labeligs g 1 ad g are super as the vertices of F are labeled with the smallest possible labels. Uder the labelig g 1 or g the path edges attai the labels +,+ 3,..., or +3,+5,...,3 1, respectively, ad the spokes admit the labels +1, +,...,3 or +,+4,...,3, respectively. For the C 3 -weights of the cycle C j 3 = v j v j+1 v +1 v j, j =1,,..., 1, we get wt g1 (C j 3 =g 1(v j +g 1 (v j v j+1 +g 1 (v j+1 +g 1 (v j+1 v +1 +g 1 (v +1 + g 1 (v j v +1 ad = ( +( + j = j j+1 +( +1 j+ ( + j+1 +( +(j +1+( +1 for 1 j 1, j 1 (mod + j +( +1 j+ j+ +( +(j +1+( +1 +( + j = j for j 1, j 0 (mod wt g (C3=g j (v j +g (v j v j+1 +g (v j+1 +g (v j+1 v +1 +g (v +1 + g (v j v +1 j +( +1+j+(j +1+( + +(j +1+( +1 +( +1+j = j for 1 j 1, j 1 (mod = j +( +1+j+(j +1+( +1+(j +1+( +1 +( + + j = j for j 1, j 0 (mod. For j =1,,..., 1thatis wt g1 (C j 3 = j ad wt g (C3=4 j j. This meas that uder the labeligs g 1 ad g the C 3 -weights form the arithmetic sequeces with the differeces ad 6, respectively. 4 C 4 -atimagicess of fas Every cycle C 4 i F is of the form C j 4 = v jv j+1 v j+ v +1 v j, j =1,,...,, ad for 4, each edge of F belogs to at least oe cycle of C j 4.FortheC 4 -weight of

9 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, the cycle C j 4, j =1,,...,, uder a total labelig f we have wt f (C j 4 =f(v j+f(v j v j+1 +f(v j+1 +f(v j+1 v j+ +f(v j+ +f(v j+ v +1 + f(v +1 +f(v j v +1. (15 From Theorems 1 through 3 it follows that F, provided ecessary coditio for the coverig by C 4 cycles is met, the there exist super (a, d-c 4 -atimagic labeligs for every d {1, 3, 5, 7, 11}. The followig theorem shows also that differeces d = 0,, 4 ad d = 6 are feasible. Theorem 5. The fa F, 4, issuper(a, d-c 4 -atimagic for d =0, 1,, 3, 4, 5, 6, 7, 11. Proof. For d =1, 3, 5, 7, 11 the results follow from Theorems 1 through 3. If d = 0,, 4, 6 let us cosider the followig. For F, 4, defie the total labeligs h m,1 t 4, i the followig way +1 i, for i =1,,..., ad m =1, 3 h m (v i = +1, for i = +1adm =1, 3 i, for i =1,,...,+1adm =, 4 h m (v i v +1 = h m (v i v i+1 = { + i, for i =1,,..., ad m =1, i, for i =1,,..., ad m =, i+1, for 1 i 1, i 1 (mod, ad m =1,, 3, i, for i 1, i 0 (mod, ad m =1,, 3, 4. It is easy to see that h m is a super labelig for every m =1,, 3, 4. Uder all labeligs the path edges attai the labels +,+3,..., ad the spokes are labeled by the labels +1, +,...,3. Accordig to (15 ( +1 j+ ( +1+ j+1 +( +1 (j +1 + ( j+1 +( +1 (j + +( +(j ++( +1+( + j = for 1 j, j 1 (mod wt h1 (C4= j ( +1 j+ ( j ( +( +1 (j (j+1+1 +( +1 (j + +( +(j ++( +1+( + j = for j, j 0 (mod,

10 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, j + ( +1+ j+1 +(j j+1 +(j + + ( + +(3 +1 (j ++( +1+(3 +1 j = j for 1 j, j 1 (mod wt h (C4= j j + ( j ( +(j (j+1+1 +(j + +(3 +1 (j ++( +1+(3 +1 j = j for j, j 0 (mod, ( +1 j+ ( +1+ j+1 +( +1 (j +1 + ( j+1 +( +1 (j + +(3 +1 (j ++( +1+(3 +1 j = j for 1 j, j 1 (mod wt h3 (C4= j ( +1 j+ ( j ( +( +1 (j (j+1+1 +( +1 (j + +(3 +1 (j ++( +1+(3 +1 j = j for j, j 0 (mod, ad j + ( +1+ j+1 +(j j+1 +(j + + ( + +( +(j ++( +1+( + j = j for 1 j, j 1 (mod wt h4 (C4= j j + ( j ( +(j (j+1+1 +(j + +( +(j ++( +1+( + j = j for j, j 0 (mod. This meas that uder the labeligs h m, m =1,, 3, 4, the C 4 -weights form the arithmetic sequeces with the differeces d = 0,, 4 ad 6, respectively.

11 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, Coclusio I this paper we examied the existece of super (a, d-c k -atimagic labeligs for fas. We proved that the fa F, 3, admits a super (a, d-c k -atimagic labelig for k =3, 4,..., + ad d {1, 3,k 7,k+1, k 5, k 1, 3k 9, 3k 1}. We showed that there exists a super (a, d-c 3 -atimagic labelig for d =0, 1,, 3, 4, 5, 6, 8 ad a super (a, d-c 4 -atimagic labelig for d =0, 1,, 3, 4, 5, 6, 7, 11 of F, 4. For further ivestigatio we propose the followig ope problem. Ope Problem 1. Fid a super (a, d-c k -atimagic labelig of the fa F for d 1, 3,k 7,k+1, k 5, k 1, 3k 9, 3k 1. Ackowledgemets The research for this article was supported by APVV Refereces [1] S. Arumugam, M. Miller, O. Phaalasy ad J. Rya, Atimagic labelig of geeralized pyramid graphs, Acta Math. Si. (Egl. Ser. 30 (014, [] M. Bača, L. Brakovic ad A. Semaičová-Feňovčíková, Labeligs of plae graphs cotaiig Hamilto path, Acta Math. Si. (Egl. Ser. 7(4 (011, [3] M. Bača, Z. Kimáková, A. Semaičová-Feňovčíková ad M. A. Umar, Treeatimagicess of discoected graphs, Math.Probl.Eg.015 (015, Article ID 50451, 4 pp. [4] M. Bača ad M. Miller, Super edge-atimagic graphs: A wealth of problems ad some solutios, Brow Walker Press, Boca Rato, Florida, 008. [5] M. Bača, M. Miller, O. Phaalasy ad A. Semaičová-Feňovčíková, Super d- atimagic labeligs of discoected plae graphs, ActaMath.Si.(Egl.Ser. 6(1 (010, [6] M. Bača, M. Miller, J. Rya ad A. Semaičová-Feňovčíková, O H-atimagicess of discoected graphs, Bull. Aust. Math. Soc. 94( (016, [7]H.Eomoto,A.S.Lladó, T. Nakamigawa ad G. Rigel, Super edge-magic graphs, SUT J. Math. 34 (1998, [8] A. Gutiérrez ad A. Lladó, Magic coverigs, J. Combi. Math. Combi. Comput. 55 (005,

12 A. OVAIS ET AL. / AUSTRALAS. J. COMBIN. 68 (1 (017, [9] N. Iayah, A. N. M. Salma ad R. Simajutak, O (a, d-h-atimagic coverigs of graphs, J. Combi. Math. Combi. Comput. 71 (009, [10] N. Iayah, R. Simajutak, A. N. M. Salma ad K. I. A. Syuhada, O (a, d- H-atimagic total labeligs for shackles of a coected graph H, Australas. J. Combi. 57 (013, [11] A. Kotzig ad A. Rosa, Magic valuatios of fiite graphs, Caad. Math. Bull. 13 (1970, [1] A. Lladó ad J. Moragas, Cycle-magic graphs, Discrete Math. 307 (007, [13] K. W. Lih, O magic ad cosecutive labeligs of plae graphs, Utilitas Math. 4 (1983, [14] A. M. Marr ad W. D. Wallis, Magic Graphs, Birkhäuser, New York, 013. [15] T. K. Maryati, A. N. M. Salma ad E. T. Baskoro, Supermagic coverigs of the disjoit uio of graphs ad amalgamatios, Discrete Math. 313 (013, [16] T. K. Maryati, A. N. M. Salma, E. T. Baskoro, J. Rya ad M. Miller, O H-supermagic labeligs for certai shackles ad amalgamatios of a coected graph, Utilitas Math. 83 (010, [17] A. A. G. Ngurah, A. N. M. Salma ad L. Susilowati, H-supermagic labeligs of graphs, Discrete Math. 310 (010, [18] A. N. M. Salma, A. A. G. Ngurah ad N. Izzati, O (super-edge-magic total labeligs of subdivisio of stars S, Utilitas Math. 81 (010, [19] A. Semaičová Feňovčíková, M. Bača, M. Lascsáková, M. Miller ad J. Rya, Wheels are cycle-atimagic, Electro. Notes Discrete Math. 48 (015, [0] R. Simajutak, M. Miller ad F. Bertault, Two ew (a, d-atimagic graph labeligs, Proc. Eleveth Australas. Workshop Combi. Alg. (AWOCA (000, (Received 16 Mar 016; revised 7 Mar 017