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

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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 ) = ;

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.

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