THE ENTIRE FACE IRREGULARITY STRENGTH OF A BOOK WITH POLYGONAL PAGES NILAI KETAKTERATURAN SELURUH MUKA GRAF BUKU SEGI BANYAK

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1 Jural Ilmu Matematika da Terapa Desember 015 Volume 9 Nomor Hal THE ENTIRE FACE IRREGULARITY STRENGTH OF A BOOK WITH POLYGONAL PAGES Meili I. Tilukay 1, Ve Y. I. Ilwaru 1, Jurusa Matematika FMIPA Uiversitas Pattimura Jl. Ir. M. Putuhea, Kampus Upatti, Poka-Ambo, Idoesia 1 meili.tilukay@fmipa.upatti.ac.id Abstract A face irregular etire labelig is itroduced by Baca et al. recetly, as a modificatio of the well-kow vertex irregular ad edge irregular total labelig of graphs ad the idea of the etire colourig of plae graph. A face irregular etire k-labelig λ: V E F {1,,, k} of a -coected plae graph G = (V, E, F) is a labelig of vertices, edges, ad faces of G such that for ay two differet faces f ad g, their weights w λ (f) ad w λ (f) are distict. The miimum k for which a plae graph G has a face irregular etire k-labelig is called the etire face irregularity stregth of G, deoted by efs(g). This paper deals with the etire face irregularity stregth of a book with m -polygoal pages, where embedded i a plae as a closed book with sided exteral face. Keywords ad phrases: Book, etire face irregularity stregth, face irregular etire k-labelig, plae graph, polygoal page. NILAI KETAKTERATURAN SELURUH MUKA GRAF BUKU SEGI BANYAK Abstrak Pelabela tak teratur seluruh muka diperkealka oleh Baca et al. baru-baru ii, sebagai suatu modifikasi atas pelabela total tak teratur titik da tak teratur sisi suatu graf serta ide tetag pewaraa legkap pada graf bidag. Pelabela k- tak teratur seluruh muka λ: V E F {1,,, k} dari suatu graf bidag -coected G = (V, E, F) adalah suatu pelabela seluruh titik, sisi, da muka iteral dari G sedemikia sehigga utuk sebarag dua muka f ad g berbeda, bobot muka w λ (f) ad w λ (f) juga berbeda. Bilaga bulat terkecil k sedemikia sehigga suatu graf bidag G memiliki suatu pelabela k-tak teratur seluruh muka disebut ilai ketakteratura seluruh muka dari G, diotasika oleh efs(g). Kami meetuka ilai eksak dari ilai ketakteratura seluruh muka graf buku segi-, dimaa pada bidag datar dapat digambarka seperti suatu buku tertutup. Kata Kuci: Graf bidag, graf buku segi-, ilai ketakteratura seluruh muka, pelabela legkap k-tak teratur muka. 1. Itroductio Let G be a fiite, simple, udirected graph with vertex set V(G)ad edge set E(G). A total labelig of G is a mappig that seds V E to a set of umbers (usually positive or oegative itegers). Accordig to the coditio defied i a total labelig, there are may types of total labelig have bee ivestigated. Baca, Jedrol, Miller, ad Rya i [1] itroduced a vertex irregular ad edge irregular total labelig of graphs. For ay total labelig f: V E {1,,, k}, the weight of a vertex v ad the weight of a edge e = xy are defied by w(v) = f(v) + uv E f(uv) ad w(xy) = f(x) + f(y) + f(xy), respectively. If all the vertex weights are distict, the f is called a vertex irregular total k-labelig, ad if all the edge weights are distict, the f is called a edge irregular total k-labelig. The miimum value of k for which there exist a vertex (a edge) irregular total labelig f: V E {1,,, k} is called the total vertex (edge) irregularity 103

2 104 Tilukay & Ilwaru The Etire Face Irregularity Stregth of a Book with Polygoal Pages stregth of G ad is deoted by tvs(g) (tes(g)), respectively. There are several bouds ad exact values of tvs ad tes were determied for differet types of graphs give i [1] ad listed i []. Furthermore, Ivaco ad Jedrol i [3] posed a cojecture that for arbitrary graph G differet from K 5 ad maximum degree (G), tes(g) = max { E(G) +, (G) + 1 }. 3 Combiig previous coditios o irregular total labelig, Marzuki et al. [4] defied a totally irregular total labelig. A total k-labelig f: V E {1,,, k} of G is called a totally irregular total k-labelig if for ay pair of vertices x ad y, their weights w(x) ad w(y) are distict ad for ay pair of edges x 1 x ad y 1 y, their weights w(x 1 x ) ad w(y 1 y ) are distict. The miimum k for which a graph G has totally irregular total labelig, is called total irregularity stregth of G, deoted by ts(g). They have proved that for every graph G, ts(g) max{tes(g), tvs(g)} (6) Several upper bouds ad exact values of ts were determied for differet types of graphs give i [4], [5], [6], ad [7]. Motivated by this graphs ivariats, Baca et al. i [8] studied irregular labelig of a plae graph by labelig vertices, edges, ad faces the cosiderig the weights of faces. They defied a face irregular etire labelig. A -coected plae graph G = (V, E, F) is a particular drawig of plaar graph o the Euclidea plae where every face is boud by a cycle.. Let G = (V, E, F) be a plae graph. A labelig λ V E F {1,,, k} is called a face irregular etire k-labelig of the plae graph G if for ay two distict faces f ad g of G, their weights w λ (f) ad w λ (f) are distict. The miimum k for which a plae graph G has a face irregular etire k-labelig is called the etire face irregularity stregth of G, deoted by efs(g). The weight of a face f uder the labelig λ is the sum of labels carried by that face ad the edges ad vertices of its boudary. They also provided the boudaries of efs(g). Teorema A. Let G = (V, E, F) be a -coected plae graph G with i i-sided faces, i 3. Let a = mi{i i 0} ad b = max{i i 0}. The a b efs(g) max{ b + 1 i 3 i b}. For b = 1, they gave the lower boud as follow Teorema B. Let G = (V, E, F) be a -coected plae graph G with i i-sided faces, i 3. Let a = mi{i i 0}, b = max{i i 0}, b = 1 ad c = max{i i 0, i < b}. The a + F 1 efs(g). c + 1 Moreover, by cosiderig the maximum degree of a -coected plae graph G, they obtaied the followig theorem. Theorem C. Let G = (V, E, F) be a -coected plae graph G with maximum degree. Let x be a vertex of degree ad let the smallest (ad biggest) face icidet with x be a a-sided (ad a b-sided) face, respectively. The efs(g) a + 1. b They proved that Theorem B is tight for Ladder graph L, 3, ad its variatio ad Theorem C is tight for wheel graph W, 3. I this paper, we determie the exact value of efs of a book with m -polygoal pages which is greater tha the lower boud give i Theorem A - C.

3 Barekeg: Jural Ilmu Matematika da Terapa Desember 015 Volume 9 Nomor Hal Mai Results Cosiderig Theorem C, efs(w ), ad a coditio where every face of a plae graph shares commo vertices or edges, our first result provide a lower boud of the etire face irregularity stregth of a graph with this coditio. This ca be cosidered as geeralizatio of Theorem A, B, ad C. Lemma.1. Let G = (V, E, F) be a -coected plae graph with i i-sided faces, i 3. Let a = mi{i i 0}, b = max{i i 0}, c = max{i i 0, i < b}, ad d be the umber of commo labels of vertices ad edges which have bouded every face of G. The a + F d 1, for c d + 1 b = 1, efs(g) a + F d { b d + 1, otherwise. Proof. Let λ V E F {1,,, k} be a face irregular etire k-labelig of -coected plae graph G = (V, E, F) with efs(g) = k. Our first proof is for b 1. By Theorem A, the miimum face-weight is at least a + 1 ad the maximum face-weight is at least a + F. Sice G is -coected, each face of G is a cycle. It implies that every face might be bouded by commo vertices ad edges. Let d be the umber of commo labels of vertices ad edges which have bouded every face of G ad D be the sum of all commo labels. The the face-weights w λ (f 1 ), w λ (f 1 ),, w λ (f F ) are all distict ad each of them cotais D, implies the variatio of face-weights is deped o a d + i b d + 1 labels. Without addig D, the maximum sum of a face label ad all vertices ad edges-labels surroudig it is at least a + F d. This is the sum of at most b d + 1 labels. Thus, we have efs(g) a+ F d b d+1. For b = 1, it is a direct cosequece from Theorem B with the same reaso as i the result above. This lower boud is tight for ladder graphs ad its variatio ad wheels give i [8]. A book with m -polygoal pages B m, m 1, 3, is a plae graph obtaied from m-copies of cycle C that share a commo edge. There are may ways drawig B m for which the exteral face of B m ca be a -sided face or a ( )-sided face. By cosiderig that topologically, B m ca be draw o a plae as a closed book such that B m has a -sided exteral face, a -sided iteral face, ad m 1 umber of ( )-sided iteral faces, the etire face irregularity stregth of B m is provided i the ext theorem. Theorem.. For B m, m 1, 3, be a book with m -polygoal pages whose a -sided exteral face, a -sided iteral face, ad m 1 ( )-sided iteral faces, we have, for m {1, }; efs(b m ) = { 4 + m 7 4 5, otherwise. Proof. Let B m, m 1, 3, be a -coected plae graph. For m {1, }, by Lemma.1, we have efs(b m ). Labelig the -sided exteral face by label ad all the rests by label 1, the all face-weights are distict. Thus, efs(b m ) =. Now for m >, let z = efs(b m ). Sice every iteral face of B m shares commo vertices, a =, b =, ad b > 1, by Lemma.1, we have z a+ F = b 1 +m 1. Cosider that z = +m 1 is ot valid, sice for m 4, the maximum label is 1. Moreover, sice B m has at least face-weights which are cotributed by the same umber of labels, there must be faces of the same weight. The the divisor must be at least 4 4. Thus we have z 4+m

4 106 Tilukay & Ilwaru The Etire Face Irregularity Stregth of a Book with Polygoal Pages Next, to show that z is a upper boud for etire face irregularity stregth of B m, let B m, m 1, 3, be the -coected plae graph with a -sided iteral face f it, m 1 ( )-sided iteral faces ad a exteral -sided face f ext. Let m 1 = m ad m = m m 1. Our goal is to have m 1 distict eve face-weights ad m distict odd face-weights such that m ( )-sided face-weights are distict ad form a arithmetic progressio. Let z = 4+m 7. It ca be see that B 4 5 m has m differet paths of legth ( 1). Next, we divide m 1 paths parts, where part s-th cosists of (4 5) paths, for 1 s S 1, ad part S-th cosists of ito S = m r 1 = m 1 (S 1)(4 5) paths. Also, we divide m paths ito T = m +1 parts, where the first part 4 5 cosists of (4 6) paths, part t-th cosists of (4 5) paths, for t T 1, ad part T-th cosists of r = m (T 1)(4 5) paths. Let V(B m ) = {x, y, u(s) i j, u(s) k j, v(t) i j v(1) 1 j, v(t) l j 1 s S 1, 1 t T 1, 1 i 4 5, 1 j, 1 k r 1, 1 l r }; E(B m ) = {xy} {u(s) i 1 = x u(s) i, u(s) i j 1 = u(s) i j u(s) i j, u(s) i 3 = u(s) i 4 y 1 s S 1, 1 i 4 5, j } {u(s) i 1 = x u(s) i, u(s) i j 1 = u(s) i j u(s) i j, u(s) i 3 = u(s) i 4 y 1 i r 1, j } {v(t) i 1 = xv(t) i, v(t) i j 1 = v(t) i j v(t) i j, v(t) i 3 = v(t) i 4 y 1 t T, 1 i 4 5, j } {v(t) i 1 = xv(t) i, v(t) i j 1 = v(t) i j v(t) i j, v(t) i 3 = v(t) i 4 y 1 i r, j }; F(B m ) = {f ext, f it, u(s) i, u(s) k, v(t) i v(1) 1, v(t) j 1 s S 1, 1 t T 1, 1 i 4 5, 1 k r 1, 1 l r }; Where f ext is bouded by cycle xv(1) v(1) 4 v(1) 4 yx; f it is bouded by cycle xu(1) 1 u(1) 1 4 u(1) 1 4 yx; u(s) i is bouded by cycle xu(s) i u(s) 4 i u(s) 4 i yu(s) 4 i+1 u(s) 6 i+1 u(s) i+1 x, for 1 s S, i r 1 ; u(s) r1 is bouded by cycle xu(s) r1 u(s) 4 r1 u(s) 4 r1 yv(t) 4 r v(t) 6 r v(t) r x; ad v(t) i is bouded by cycle xv(t) i v(t) 4 i v(t) 4 i yv(t) 4 i+1 v(t) 6 i+1 v(t) i+1 x, for 1 t T, i r ; Our otatios above imply that, without losig geerality, for v(t) i j, we let i 4 5 for t = 1. It meas that there is o vertex or edge or face v(1) 1 j. Now, we divide our labelig of B m ito cases as follows: Case 1. For odd m with r 1 or eve m; Defie a etire k-labelig λ V E F {1,,, k} of B m as follows. λ(x) = λ(y) = λ(xy) = λ(f ext ) = 1; λ(f it ) = ;

5 Barekeg: Jural Ilmu Matematika da Terapa Desember 015 Volume 9 Nomor Hal λ(u(s) i j ) = λ (v(t) i j ) = s 1 for 1 s S, 1 i mi{r 1, } ad 1 j i 1 s for 1 s S, 1 i mi{r 1, } ad i j s for 1 s S, 1 i mi{r 1, 4 5} ad 1 j i + { s + 1 for 1 s S, 1 i mi{r 1, 4 5} ad i + 1 j t 1, for 1 t T, 1 i mi{r, } ad 1 j i ; t, for 1 t T, 1 i mi{r, } ad i 1 j 3; t, for 1 t T, 1 i mi{r, 4 5} ad 1 j i + 3; t + 1, for 1 t T, 1 i mi{r, 4 5} ad i + j 3; t, for 1 t T, i = 1 ad j = ; t 1, for 1 t T, i mi{r, 1} ad j = ; t, for 1 t T 1, i 4 5 ad j =. { t, for t = T, 1 i mi{r 1, 4 6} ad j = Case. For odd m with r = 1 or r 4 5; Defie a etire k-labelig λ V E F {1,,, k} of B m as follows. λ (x) = λ (y) = λ (xy) = λ (f ext ) = 1; λ (f it ) = ; λ (u(s) i j ) = λ(u(s) i j ) λ (v(t) i j ) = T, for r = 1, t = T, i = 1, j = 1; T 1, for r = 1, t = T 1, i = 4 5, j = ; λ(v(t) i j ) + 1, for r odd, r 4 5, t = T, i = r, j = 1; λ(v(t) i j ) 1, for r odd, r 4 5, t = T, i = r 1, j = ; λ(v(t) i j ) 1, for r eve, r 4 5, t = T, i = r 1, j = 3; λ(v(t) i j ) + 1, for r eve, r 4 5, t = T, i = r 1, j = ; { λ(v(t) j i ), for otherwise. It is easy to check that the labelig λ is a etire z-labelig. The we have evaluate the face weights set {w(f ext ), w(f it ), w(u(s) i ), w(v(t) i ) 1 s S, 1 t T, 1 i 4 5} as follows. w(f ext ) = + 1; w(f it ) = + ; (s 1)(4 5) + i, for 1 s S 1, 1 i 4 5; w(u(s) (s 1)(4 5) + i, for s = S 1, 1 i r i ) = 1 ; (s 1)(4 5) + r 1, for eve m, s = S 1, i = r 1 ; { (s 1)(4 5) + r 1 1, for odd m, s = S 1, i = r 1. w(v(t) (t 1)(4 5) + i + 1, for 1 t T 1, 1 i 4 5; i ) = { (T 1)(4 5) + i + 1, for t = T, 1 i r 1. Sice all face-weights are distict, the λ is a face irregular etire z-labelig of B m where m is odd with r 1 or m is eve; ad λ is a face irregular etire z-labelig of B m where m is odd with r = 1 or r 4 5. Thus, z = 4+m 7 is the etire face irregularity stregth of B 4 5 m. Note that our result i Theorem. show that the efs(b m ) is greater tha the lower boud i Lemma.1. Hece, we propose the followig ope problem.

6 108 Tilukay & Ilwaru The Etire Face Irregularity Stregth of a Book with Polygoal Pages Ope Problems 1. Fid a class of graph which satisfy a coditio where the lower boud i Lemma.1 is sharp;. Geeralize the lower boud for ay coditio. Refereces [1] M. Baca, S. Jedrol, M. Miller ad J. Rya, O Irregular Total Labeligs, Discrete Mathematics, vol. 307, pp , 007. [] J. A. Galia, A Dyamic Survey of Graph Labelig, Electroic Joural of Combiatorics, vol. 18 #DS6, 015. [3] J. Ivaco ad S. Jedrol, The Total Edge Irregularity Stregth of Trees, Discuss. Math. Graph Theory, vol. 6, pp , 006. [4] C. C. Marzuki, A. N. M. Salma ad M. Miller, O The Total Irregularity Stregths of Cycles ad Paths, Far East Joural of Mathematical Scieces, vol. 8 (1), pp. 1-1, 013. [5] R. Ramdai ad A. N. M. Salma, O The Total Irregularity Stregths of Some Cartesia Products Graphs, AKCE It. J. Graphs Comb., vol. 10 No., pp , 013. [6] R. Ramdai, A. N. M. Salma, H. Assiyatu, A. Semaicova-Feovcikova ad M. Baca, Total Irregularity Stregth of Three Family of Graphs, Math. Comput. Sci, vol. 9, pp. 9-37, 015. [7] M. I. Tilukay, A. N. M. Salma ad E. R. Persulessy, "O The Total Irregularity Stregth of Fa, Wheel, Triagular Book, ad Friedship Graphs," Procedia Computer Sciece, vol. 74, pp , 015. [8] M. Baca, S. Jedrol, K. Kathiresa ad K. Muthugurupackiam, Etire Labelig of Plae Graphs, Applied Mathematics ad Iformatio Scieces, vol. 9, o. 1, pp , 015.