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, PokaAmbo, 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 wellkow vertex irregular ad edge irregular total labelig of graphs ad the idea of the etire colourig of plae graph. A face irregular etire klabelig 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 klabelig 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 klabelig, plae graph, polygoal page. NILAI KETAKTERATURAN SELURUH MUKA GRAF BUKU SEGI BANYAK Abstrak Pelabela tak teratur seluruh muka diperkealka oleh Baca et al. barubaru 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 ktak 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 ktak 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 klabelig, ad if all the edge weights are distict, the f is called a edge irregular total klabelig. 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 klabelig f: V E {1,,, k} of G is called a totally irregular total klabelig 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 klabelig 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 klabelig 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 isided 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 isided 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 asided (ad a bsided) 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 isided 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 klabelig of coected plae graph G = (V, E, F) with efs(g) = k. Our first proof is for b 1. By Theorem A, the miimum faceweight is at least a + 1 ad the maximum faceweight 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 faceweights 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 faceweights is deped o a d + i b d + 1 labels. Without addig D, the maximum sum of a face label ad all vertices ad edgeslabels 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 mcopies 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 faceweights 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 faceweights 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 faceweights ad m distict odd faceweights such that m ( )sided faceweights 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 sth cosists of (4 5) paths, for 1 s S 1, ad part Sth 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 tth cosists of (4 5) paths, for t T 1, ad part Tth 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 klabelig 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 klabelig 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 zlabelig. 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 faceweights are distict, the λ is a face irregular etire zlabelig of B m where m is odd with r 1 or m is eve; ad λ is a face irregular etire zlabelig 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. 11, 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. SemaicovaFeovcikova ad M. Baca, Total Irregularity Stregth of Three Family of Graphs, Math. Comput. Sci, vol. 9, pp. 937, 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.
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