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THE 3RD INTERNATIONAL CONFERENCE ON MATHEMATICS AND STATISTICS BOGOR, 5 – 6 AUGUST 2008 Mathematics and Statistics: bridge for academia, business, and government in the entrepreneurial era

organized by

MSMSSEA (Moslems Statisticians and Mathematicians Society in South East Asia)

Department of Statistics Department of Mathematics Institut Pertanian Bogor

Department of Mathematics Universiti Malaysia Terengganu, Malaysia


Assalaamu’alaikum warahmatullaahi wabarakaatuh Welcome all participants of ICoMS 2008 to Bogor – Indonesia. This event is organized by MSMSSEA in collaboration with Institut Pertanian Bogor (Indonesia) and Universiti Malaysia Terenganu (Malaysia). We, the organizing committee, are very glad having this international conference due to many reasons. 1. ICoMS is a good avenue for mathematicians, statisticians, and other scientist to communicate. 2. ICoMS 2008 has a theme related to entrepreneurial era which is very important for mathematicians and statisticians, and scientist in general. 3. The event is important venue for business group, government, and academia to communicate and share knowledge as well. 4. Bogor is beautiful place in Indonesia surrounded by many research centers, IPB, Botanical garden, an other point of interest related to research institution. We are also happy that the Vice President of Republic of Indonesia, Ministry of National Education, Ministry of Energy and Mineral Resources, and Ministry of Communication and Information Technology are supporting to the ICoMS 2008. This event held on two days, August 5-6, and consist of several parts. We invite 17 outstanding professors to share and discuss topics in mathematics and statistics, including application. As many as 170 paper and 30 posters presented during this twoday conference. We appreciate to all of contributor from various countries who are motivated to participate in this event. High appreciation is also awarded to companies and agencies which facilitate so that the even could run well. We really hope all participants can benefit many things from this international event. May God bless you.

Wa’alaikumsalam warahmatullaahi wabarakaatuh. The Committee of ICoMS 2008


CONTENTS Preface Contents Speech of Director General of Higher Education: Dr. Fasli Jalal SESSION A Vector-Borne Disease Transmission Model: The Case of Filariasis Transmission in Jati Sampurna Optimization of Dining Table Placement in Restaurant Using Genetic Algorithm

ii iii xiii

A.K. Supriatna a;¤ H. Serviana b E. Soewono


Monika, Arnold Aribowo, Samuel Lukas


The Influence of Islamic Mathematics In The Malay World: A Study of Classical Astronomical Tables

Baharrudin Zainal and Mat Rofa Ismail


Problem-Based Learning Strategy To Improve Mathematics Skills Among Students

Ismail Abdullah


Linear Time Invariant Versus Linear Parameter Varying Reduced Order Controller Design For Aircraft Model



Scientific Computing Research In The Faculty of Information Science And Technology Universiti Kebangsaan Malaysia

Mohammad Khatim Hasan, Muriati Mokhtar, Riza Sulaiman and Jumat Sulaiman


The Functions of Contextual Problems On A Mathematics Instruction Implementing The Realistic Approach

Rini Setianingsih


Further Results On The Ramsey Numbers For Star Union Cycle Versus Wheel On Seven Vertices

I Wayan Sudarsana, Edy Tri Baskoro, and Hilda Assiyatun


Characteristics of Lecturer That Influences Effective Teaching In Quality Education Based On Tqm Concepts

Khairul Anuar Mohd Ali, Zainol Mustafa, Fazli Idris & Siau Yee


Feynman Diagrams And The Tau Function

Zainal Abdul Aziz


An Evaluation Of The Efficiency Wage Model: Evidence From Bangladesh Food Industry

Masud Rana, Md. Azizul Baten and Mezbahur Rahman


Modeling Self-Potential (Sp) Distribution Groundwater Flow

Muhammad Hamzah, S., Santoso, D., Parnadi, W.W., Sulistijo, B


Nonlinear Dynamic Plant Identification Of Waste Heat Boiler Unit Using Adaptive Wavelet - Neural Network (Wavenet) Method



A Group Mutual Exclusion Algorithm For Ad Hoc Mobile Networks

Armin Lawi


On The Measurement Of Credit Risk: A New Geometric Approach

Alireza Bahiraie, Noor Akma Ibrahim, Ismail bin Mohd, Azhar A.K.M


Using Boundary Element Method (BEM)


Slide By Slide Method In Solving General Global Optimization Problems

Goh Khang Wen, Ismail bin Mohd, and Yosza bin Dasril


The Infinite Divisibility Of Geometric Distribution

Dodi Devianto and Katsuo Takano


Tight Immersions And Transnormal Embeddings



Hydromagnetic Flow Over An Impermeable Linearly Stretching Sheet Immersed In A Non-Darcian Porous Medium

Noor Fadiya Mohd Noor and Ishak Hashim


Wavelet Analysis Of Solar Mass Ejection During Active And Quiet Sun

Saifuddin Ahmed Jilani and M.Ayub Khan YousfZai


Modeling A Determination Of Expected Commercial Value Of A Project In Producing Innovative Products In Smes

Nursafarizah Abd Aziz, Nor Ratna Masrom, Yosza Bin Dasril, and Adi Saptari


Modelling Nurse Rostering Using A 0-1 Goal Programming: A Case Study In Hukm

Ruzzakiah Jenal, Wan Rosmanira Ismail, Liong Choong Yeun And Masri Binti Ayob


Enhancing Students’ Mathematical Learning through Teacher Professional Development Around Prime And Maximal Ideals Of A Skew Polynomial Ring Over A Dedekind Domain

Yaya S. Kusumah


A. K. Amir, P. Astuti, and I. Muchtadi-Alamsyah


The Fuzzy Version Of The Fundamental Theorem Of Semigroup Homomorphism

Karyati , Indah Emilia W, Sri Wahyuni, Budi Surodjo, Setiadji


Solving Fourth-Order Parabolic Equations By Red-Black Quarter-Sweep Sor Iterative Methods

J. Sulaiman, M. Othman, and M.K. Hasan


25 Years Development Of Knowledge Graph Theory: The Results And The Challenge

Sri Nurdiati and Cornelis Hoede


Eigenvalues And Eigenvectors Of Matrices Over Fuzzy Number Max-Plus Algebra

M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, and F. Susilo


The Partition Dimension Of Windmill Graph

Darmaji, Novian Syah, Saladin Uttunggadewa, Edy Tri Baskoro


Sorting Process Of Two Sets Of Non-Quantitative Data: Mathematical Method And Analysis

Edi Cahyono, David Taniar, La Ode Saidi, Arman and Natalis Ransi


On The Metric Dimension Of P2[Pn]

S. Widosaputro, E.T. Baskoro, A.N.M. Salman, and D. Suprijanto


IS-LM In Slow-Fast System

Joice Ruth Juliana, Endah Asmawati




A Four-Stage Fifth-Order Runge-Kutta-Nystrom Methods With Dispersion Of High Order

Norazak Senu, Mohamed Suleiman, Fudziah Ismail, Mohamed Othman, and Norfifah [email protected]


Logic In Recurrent Hopfield Network

Saratha Sathasivam


The Generalization Of Incidence Algebra

Ema Carnia, Sri Wahyuni, Irawati and Setiadji


Representation Of Sm-Operators On Product Spaces

Muslim Ansori ,Soeparna Darmawijaya and Supama


Benard-Marangoni Instability In A Rotating Fluid Layer With Feedback Control Strategy

Zailan Siri and Ishak Hashim


Interior Point Methods For Solving Linear Programing

Iwan Tri Riyadi Yanto,Julan Hernadi, and Yudi Ari Adi


Endo-Prime N-Group

Indah Emilia Wijayanti


Modelling Of Budget Allocation For University Library

Engku Muhammad Nazri Bin Engku Abu Bakar, Syariza Abdul Rahman, Noorezatty Mohd Yusop


A Novel Natural Approach To Euclidean TSP

Nur Azman Abu, Shahrin Sahib And Nanna Suryana


R-Linear Independent Generalization

Suprapto, Sri Wahyuni, Indah Emilia W., Irawati


A Study For Hyperbolic Decline Exponent Bound

S. Wahyuningsih, S. Darwis, A.Y. Gunawan, and A.K. Permadi


The F-Coloring Of The Corona Product Of

Adiwijaya, A.N.M. Salman, E.T. Baskoro, and D. Suprijanto


Shukhrat I. Rakhimov, Mohamed Othman


Asymptotic Behavior Of Linear Delay Differential Systems

Eti Dwi Wiraningsih, Widodo, Lina Aryati, Syamsuddin Toaha


On total vertex-irregular labellings of tPn a forest constructed from a disjoint union of paths

Nurdin1, E.T. Baskoro, A.N.M. Salman, N.N. Gaos



r − Lebesgue Spaces Lp ( E , Σ, µ ) , 1 < p < ∞

Complete Graph With Cycle Graph

A Modified Explicit Group Iterative Algorithm With Accelerated Over-Relaxation For Solving Poisson Equation


SESSION C The Effectiveness Of The Contextual Video As A Teaching Tool In The Teaching And Learning Statistics At The Universiti Tun Hussein Onn Malaysia (Uthm)

Nafisah Kamariah Md Kamaruddin, Zulkarnain Md Amin, Norfadzilah Ishak, Wan Mohd Rashid Wan Ahmad, And Maizam Alias


The Nonlinear Mechanism Of Tsunami Wave Generation

Nazeeruddin Yaacob And Zainal Abdul Aziz


Open Economy Macroeconomics:

Iman Sugema and Toni Bakhtiar


The Range Of Mathematical Capability Of University Math Lecturers In Jakarta Indonesia

Ramir Santos Austria


An Application Wavelet Based Preconditioner For The Solution Of Ordinary Differential Equation

Ismail Bin Mohd and Farikhin


Research And Statistic Studying Model At Junior High School By Realistic Mathematics Education Application

Yani Ramdani


Using The Algebra Of Hypergraph For Reconstruction Phylogenetic Trees

Mulia Astuti, Irawati, Intan Muchtadi-Alamsyah,Ahmad Muchlis, Achirul Akbar Dan Muliana. A. Halim


Interval Linear Programming

Herry Suprajitno, Ismail Bin Mohd


What Happen With Numbers At Our Primary School?

Mohini Mohamed, Zulkifli, Jasmaniah


Determination Of GPS Signal PathUsing The Runge-Kutta Method

Mardina Abdullah, Siti Sarah Nik Zulkifli, Mahamod Ismail, Ahmad Mahir Razali And Azami Zaharim


A Novel Linguistic Aggregation Method For Group Decision Making

Zamali Tarmudi, Mohd Lazim Abdullah And Abu Osman Md Tap


A Series Of Element Shape Functions For Infinite Elements

Sri Mardiyati


Promoting Creavity In Learning Mathematics Using Open-Ended Problems

Tatag Yuli Eko Siswono


Modeling Traffic Lights In Intersection

Dieky Adzkiya And Subiono


An Alternative Formulation For Electromagnetic Wave Propagations In Source Free Region

Noraini Md Nusi And Mohamed Othman


On The Number Of Families Of Branching Processes With Immigration With Family Sizes

Husna Hasan


A Linear Rational Expectations Model

Using Petri Nets


Marangoni Convection In A Fluid Layer With Non-Uniform Temperature Gradient

Norihan Md. Arifin, Siti Suzilliana Putri Mohamed Isa, Roslinda Mohd Nazar, and Mohd Noor Saad


Dubrovin Valuation Rings Of Skew Ploynomial Rings

Intan Muchtadi Alamsyah


The Roles Of Numerical Method And Optimal Control Theory In Cancer Immunotherapy

Ismail Bin Mohd, Arif Bin Mandangan


The Development of Applied Statistical Analysis

H. Ahmad Ansori Mattjik


Risk Factors For Water Fowl Infection With Avian Influenza H5n1, West Java Province, Indonesia

Etih Sudarnika, Asep Saefuddin, Abdul Zahid And Chaerul Basri


Trend Of The Import And Export Of Cocoa In Malaysia

H. J. Zainodin, G. Khuneswari & S.C. Albert Ling


Performance Analysis Of Reactive Mobile Ad Hoc Networks Routing Protocols Based On Taguchi Technique

Hazura Mohamed, Muhammad Hisyam Lee, Mazalan Sarahintu,



Shaharuddin Salleh, And Bahrom Sanugi Optimum Designs Of Multiresponse Surface Models For The First Order Lattice Simplex Designs

Ruslan , Susanti L, Purhadi, Sony S


Bayesian Approach For Choice-Conjoint Model In Consumer Preferences

Zulhanif, Ismail Bin Mohd, Noor Akma Ibrahim, Mustafa Bin Mamat


Simultan Equation Models Of Gross Domestic Product By Exchange Rate And Money Supply Scenario Analysis For Indonesian Economy

Bagus Sumargo


On Designing Algorithm For Sample Selection

L. Muhamad Safiih And Yaya Sudarya Triana


Generating Claim Data Of General Insurance Based On Collective Risk Model And Claim Process

Aceng K. Mutaqin, Dumaria R. Tampubolon, Sutawanir Darwis


An Application On Multiple-Correspondence Analysis On The Survey For Implementation Of The Profit-Loss Sharing Concept

Novriana Sumarti, Nurdinintya Athari S., And M. Rizka Fadhli


GIS Spatial Data Visualization Tools For Artificial Reefs Distribution

Mustafa Man, Md Yazid Mohd Saman, Noor Maizura M. Noor


Khalid Samo And W.Aezwani W.A.Bakar


An Alternative Approach In Getting A Representative Model In A Mutiple Regression Analysis

G. Khuneswari, H. J. Zainodin, G. Darmesah & S. H. Sim


Wireless Internet Usage Among Students In Universiti Malaysia Sabah

Sathissan Ragavan, Darmesah Gabda, Amran Ahmed


An evaluation of a software for circular variables: ORIANA

Siti Fatimah Hassan, Abdul Ghapor Hussin and Yong Zulina Zubairi


Evaluation Of Some Methods For Estimating Parameters Of Regression Model With Various Zero Observations By Monte Carlo Simulation

Fitria Virgantari, Tjut Awaliyah, I Wayan Mangku, and Siswadi


Stability Model’s By “Eberhart-Russel’s” And Biplot Of Qpm (Quality Protein Maize) Under Central Maize In Indonesia

M Yasin Hg., Sigit Budi Santoso., And Sri Sunarti


Increasing Power Of Robust Test Through Pre-Testing In Multivariate Simple Regression Model

Rossita M. Yunus And Shahjahan Khan


Application Of Buhlmann-Straub Model On Tectonic Earthquake Insurance Problem

Hasih Pratiwi, Subanar, Danardono, And J.A.M. Van Der Weide


Evaluating The Cox-Aalen Model



Estimation Of Distributed Lag Model With Adaptive Ekspectation And Partial Adjustment On The Distribution Of Fmcd-Based Robust Mahalanobis Distance

Aidawayati Rangkuti


Hazlina Hj Ali, Maman A. Djauhari,and Sharipah S.S. Yahaya


Organizational Commitment Of Public Sector Employees In Pakistan: A Statistical Evidence

Dr. M. Rashid Salahria and Qasim Zafar


Carbon Based Material Processing Method Using Statistical Application Technique

N. Hashim, A.N. Zainal Abidin, M. Deraman, W. R. Wan Abdullah, A. Mohd Ramli, 6R.M. Yunus


Continuous Time Model For Portfolio Problem Using Dynamic Programming Approach

Sugiyarto, Ismail Mohd., Mustafa Mamat And Yosza Dasril


A Comparison of MLE and GEE On Modeling Binary Panel Response

Jaka Nugraha, Suryo Guritno, Sri Haryatmi


Bayesian Scan Statistic For Spatial Cluster Detection

Setia Gunawan Wijaya, Dian Lestari, And Yekti Widyaningsih


Correlation Coefficient Estimation From Grouped Data

Teti Sofia Yanti


The Performance Of Exponential Weighted Moving Average (Ewma) Control Chart Between Classical, Robust And Bootstrap Method

Khalida Binti Oseman, Nazaruddin Omar And Habshah Midi




Inventory Model With Gamma Distribution

Hadi Sumadibrata, Ismail Bin Mohd


Accuracy Analysis Of Naive Bayesian

Ruslam, Armin Lawi, And Sri Astuti Thamrin


A New Method For Generating Fuzzy Rules From Training Data And Its Application In Financial Problems

Agus Maman Abadi, Subanar, Widodo, Samsubar Saleh


The Application Of Laws Of Large Numbers In Convergence Concept In Probability And Distribution

Georgina M. Tinungki


An Empirical Bayes Approach for Binary Response Data in Small Area Estimation

Dian Handayani, Noor Akma Ibrahim, Khairil A. Notodiputro, MOhd. Bakri Adam


Statistical Models For Small Area Estimation

Khairil A Notodiputro, Anang Kurnia, and Kusman Sadik


Maximum Likelihood Estimation For The Non-Separable Spatial Unilateral Autoregressive Model

Norhashidah Awang, Mahendran Shitan


Small Area Estimation Using Natural Exponential Families With Quadratic Variance Function (Nef-Qvf) For Binary Data



Using An Extended And Ensemble Kalman Filter Algorithm For The Training Of Feedforward Neural Network In Time Series Forecasting

Zaqiatud Darojah, M. Isa Irawan, And Erna Apriliani


Estimation Of Outstanding Claims Liability And Sensitivity Analysis: Probabilistic Trend Family (PTF) Model

Arif Herlambang, Dumaria R Tampubolon


Expected Value Of Shot Noise Processes



Modelling Malaysian Wind Speed Data Via Two Paramaters Weibull

Nur Arina Basilah Kamisan, Yong Zulina Zubairi, Abdul Ghapor Hussin, Mohd. Sahar Yahya


Application Of Latin Hypercube Sampling And Monte Carlo Simulation Methods: Case Study The Reliability Of Stress Intensity Factor And Energy Release Rate Of Indonesian Hardwoods

Yosafat Aji Pranata And Pricillia Sofyan Tanuwijaya


The Development Of Markov Chain Monte Carlo (Mcmc) Algorithm For Autologistic Regression Parameters Estimation

Suci Astutik, Rahma Fitriani, Umu Sa’adah, And Agustin Iskandar


A Note About Dh-Fever Estimation With ARIMAX Models

Elly Ana, Dwi Atmono Agus W


Evaluation Of Additive-Innovational Outlier Identification Procedure For Some Bilinear Models



Anti-Spam Filter


smail, M.I., Mohamed, I.B., Yahya, M.S.

Interval Estimation For Quantile On One Parameter Exponential Distribution Under Multiple Type-Ii Censoring On Complex Case

Akhmad Fauzy


A Detection Measure Of Influential Observation Based On Forward Search Approach For Cox-Regression

Mohamed, I. B., Noh, N. A. M., Taib, N. A. M.


Gee-Smoothing Spline For Longitudinal Data

Suliadi, Noor Akma Ibrahim, Isthrinayagy S. Krishnarajah, and Isa Daud


On Some Theory And Applications Of Bayesian Hierarchical Modeling

Kamarulzaman Ibrahim


Simultaneous Estimation After Selection And Ranking And Other Procedures :

Suryo Guritno


Survival Probabilities Of Genes In Partial Selfing Populations

Muhamad Sabran


The Probability Difference Indices And Empirical Sampling Distribution For Dif Indices For Identifying Item Bias In Multidimensional Item Response Theory

Badrun Kartowagiran And Heri Retnawati


Volumetric Stem Biomass: A Comparitive Study Using Multiple Regression Models

Noraini Abdullah, Zainodin Hj. Jubok And Amran Ahmed


An Innovative Approach In Analysing Wind Data Via Graphical Display

Fakhrulrozi Hussain, Yong Zulina Zubairi, And Abdul Ghapor Hussin


Denoising Time Series Data Using Daubechies Wavelet Packet Transformation

Samsul Ariffin Abdul Karim, Mohd Tahir Ismail


Detection Of Outliers In Circular Regression Model Via Row Deletion Approach

Abuzaid, A. H., Mohamed, I. B., And Hussin, A.G.


Implementation Of Classification Predictive Association Rule

Herwanto, Imas S. Sitanggang


Bayesian Survival Analysis Of Acute Leukemia Patients Using Multivariate Adaptive Regression Spline Model

Nurhayati Ulath, Sri Astuti Thamrin, And Armin Lawi


Combining Individual Learning And Group Discussion In Calculus Course

Endah Asmawati, And Joice Ruth Juliana


PBSTAT: A Web-Based Statistical Analysis Software

Willy Bayuardi Suwarno, Sobir, Hajrial Aswidinnoor, And Muhamad Syukur


Reliability Of The Specific Gravity (Sg) Value Of Three Indonesian Hardwoods Using Experimental Test And Monte Carlo Simulation

Yosafat Aji Pranata And Pricillia Sofyan Tanuwijaya


Comparison Of Differencing Parameter Estimation From Arfima

Gumgum Darmawan, Nur



The Negative Exponential Case

( CPAR) Algorithm To Diabetes Diagnose

For Participatory Plant Breeding


Model By Spectral Regression Methods

Iriawan, Suhartono

Application Of Cluster Analysis To Developing Core Collection In Plant Genetic Resources



Small Area Estimation With Time And Area Effects Using A Dynamic Linear Model

Kusman Sadik And Khairil Anwar Notodiputro


Statistical Analysis Of Wind Direction Data

Ahmad Mahir Razali, Arfah Ahmad, Azami Zaharim And Kamaruzzaman Sopian


Generalized Additive Mixed Models in Small Area Estimation

Anang Kurnia, Khairil A. Notodiputro, Asep Saefuddin, I Wayan Mangku


Kernel Principal Component Analysis In Data Visualization

Ismail Djakaria, Suryo Guritno, Sri Haryatmi


GARCH Models And The Simulations

Nelson Nainggolan, Budi Nurani Ruchjana And Sutawanir Darwis


Rainfall Prediction Using Bayesian Network

Hera Faizal Rachmat, Aji Hamim Wigena, and Erfiani


Identifying Item Bias Using The Simple Volume Indices And Multidimensional Item Response Theory Likelihood Ratio (Irt-Lr) Test

Heri Retnawati


Ordinary Kriging And Inverse Distance Weighting For Mapping Soil Phosphorus In Paddy Field

Mohammad Masjkur, Muhammad Nuraidi and Chichi Noviant


K-Means Clustering Visualization On Agriculture Potential Data For Villages In Bogor Using Mapserver

Imas S. Sitanggang, Henri Harianja, and Lailan Syaufina


Some Methods To Estimate The Number Of Components In A Mixture

M. A. Satyawan, A. H. Wigena, Erfiani


A Probabilistic Model For Finding A Repeat Triplet Region In DNA Sequence Application Of Spherical Harmonics In Determination Of Tec Using Gps Observable

Tigor Nauli


Mardina Abdullah, Siti Aminah Bahari, Baharudin Yatim, Azami Zaharim, Ahmad Mahir Razali


Testing Structure Correlation Of Global Market By Statistic Vvsv

Erna Tri Herdiani, and Maman A. Djauhari


Exploring the MAUP from a spatial perspective

Gandhi Pawitan


Estimation of RCA(1) Model using EF: A new procedure and its robustness

1Norli Anida Abdullah, 2Ibrahim Mohamed, 3Shelton Peiris


Second Order Linear Elliptic Operators In The Unit Square

Abdul Rouf Alghofari



POSTER Study Of Fractional Factorial Split-Plot Experiment

Sri Winarni, Budi Susetyo, and Bagus Sartono


Improving Model Performance For Predicting Poverty Village Category Using Neighborhood Information In Bogor

Bagus Sartono, Bagus Sartono, and Zulhelmi Thaib


Ammi Models On Count Data: Log-Bilinear Models

Alfian Futuhul Hadi H. Ahmad Ansori Mattjik I Made Sumertajaya Halimatus Sa’diyah Budi Nurani R , and Kartlos J. Kachiashvili


Atje Setiawan A. , Retantyo Wardoyo , Sri Hartati , and Agus Harjoko


Validation Of Training Model For Robust Tests Of Spread

Teh Sin Yin, and Abdul Rahman Othman


Spectral Approach For Time Series Analysis

Kusman Sadik


The ACE Algorithm for Optimal Transformations in Multiple Regression The Relation Between The Students’ Interaction And The Construction Of Mathematical Knowledge Application of Auto Logistic Regression Spatial Model using Variogram Based Weighting Matrix to Predict Poverty Village Category

Kusman Sadik


Rini Setianingsih


Utami Dyah Syafitri, Bagus Sartono, Vinda Pratama


Prediction Of Oil Production Using Non Linear Regression By Sdpro Software (Special Program Package)*) An Implementation Of Spatial Data Mining Using Spatial Autoregressive (Sar) Model For Education Quality Mapping At West Java*)



The 3rd International Conference on Mathematics and Statistics (ICoMS-3) Institut Pertanian Bogor, Indonesia, 5-6 August 2008


Adiwijaya*, 2A.N.M. Salman, 3E.T. Baskoro, and 4D. Suprijanto 1,2,3,4

Combinatorial Mathematics Research Division Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung Jl. Ganesa 10 Bandung, Indonesia 40132 *

Applied Mathematics Research Group Faculty of Science, Institut Teknologi Telkom Jl. Telekomunikasi no.1 Dayeuh Kolot - Bandung, Indonesia 40257 Email : [email protected], {2msalman, 3ebaskoro, 4djoko} ∗

Abstract. Let G = (V, E) be a simple graph and f be a function which assigns a positive integer f (v) to each vertex v ∈ V. An f-coloring of graph G is a generalized edge-coloring such that each vertex v ∈ V has at most f(v) edges colored with the same color. The minimum number of colors needed to f-color G is called an f-chromatic index of G, and denoted by χf’(G). The f–chromatic index of G is equal to ∆f (G) or ∆f (G)+1, where ∆f (G) = maxv{d(v)/f(v)}. G is called in the class-1, denoted by Cf1, if χf‘(G) = ∆f (G); otherwise G is called in the class-2, denoted by Cf 2. In this paper, we assume that f (v) = k for every v ∈ V(G). We showed that for every n,m ∈ Ζ+, the corona product of Kn with Cm to be in Cf 1. Keywords : edge coloring, f-coloring, f-chromatic index, classification, corona product

1. Introduction Let G = (V(G), E(G)) be a simple graph with vertex set V(G) and edge set E (G). For each v∈ V (G), let d(v) denote the degree of v. In the classical edge coloring, every two adjacent edges gets two diferent colors. Hakimi and Kariv [3] generalized the classical edge coloring. Let f be a function from V(G) to a set of positive integers. An f -coloring of G is a coloring of edges such that each vertex v has at most f (v) edges colored with a same color. The minimum number of colors needed to f-color G is called an f-chromatic index of G, denoted by χf’(G). If f (v) = 1 for every v ∈ V (G ) , the f-coloring problem is reduced to the classical edge-coloring problem. The f-coloring problem is to find χf’(G) of a given graph G. It arises in many applications, including the network design problem, scheduling problem, the file transfer problem in a computer network [1,2]. The file transfer problem in a computer network is modelled as follows. Each computer is represented by vertex and every two computer in the file transfer process is represented by edge. Each computer v has a limit number f(v) of communications ports and it takes an equal amount of time to transfer each file. Under this assumptions, the scheduling to minimize the total time for the overall transfer process corresponds to an f-coloring of a graph G, with minimum number of colors. Note that the edges colored with the same color correspond to files that can be transferred simultaneously. Let G be a graph,   d ( v )   (1) ∆ f ( G ) = max    v∈V ( G ) f v ( )     where d(v) is the degree of vertex v in G. Hakimi and Kariv [3] gave the lower and upper bound of f-chromatic index of any graph G as follows : (2) ∆ f ( G ) ≤ χ ' f ( G ) ≤ ∆ f ( G ) + 1. ∗

G is called in the class-1, denoted by Cf1, if χf‘(G) = ∆f (G); otherwise G is called in the class-2, denoted by Cf2. Moreover, Hakimi and Kariv [1] showed that any bipartite graph is in Cf 1. If G is a graph with f(v) is even for each v∈V(G), then G must be in Cf1. Nakano et al. in [5] gave an upper bound of


f-chromatic index for a multigraph. Zhou et al. in [11] gave an algorithm for f –coloring for bipartite graphs and planar graphs. Jiguo Yu et al. [6] studied f-coloring in fans and wheels. They gave sufficient conditions for fans and wheels graph being Cf 1. Let,   d (v ) (3) V0* ( G ) = v | ∆ f ( G ) = , v ∈V ( G )  f ( v )  

   d ( v)  V * (G) = v | ∆ f (G) =  , v ∈V (G)    f ( v)    Zhang and Liu in [7,8] presented some sufficient conditions for a graph being Cf 1 as follows. Teorema 1. [7] Let G be a graph. If f (v* ) Œd (v* ) for every Teorema 2. [8] Let G be a graph. If the subgraph induced by

∗ ∗


, then G ∈ Cf 1.

is forest, then G ∈ Cf 1.

Meanwhile, Zhang and Liu [8] found the f-chromatic index for complete graphs and they gave a classification of complete graphs on f-coloring. They showed that if k and n are odd integers, f(v) = k, and k | d(v) for all v∈V, then G is of Cf 2. Otherwise G is of Cf 1. In [10], Zhang et al. presented a classification of regular graphs on f-coloring. They gave some sufficient conditions for a regular graph being Cf 1 or Cf 2. Let G dan H be two graph with n and m vertices, respectively. The corona product of graphs G and H, denoted by G  H , is defined as a graph obtained by taking one copy of G and n copies of H (say H1, H2, ..., Hn), and joining the i-th vertex of G to every vertices in Hi. Let Kn be a complete graph with n vertices and Cm be a cycle graph with m vertices. We give In the figure 1, an example of the corona product of K4 with C3.

Figure 1. The graph of corona product,

K 4  C3

However, characterization of all graphs in Cf 1 is still not completely solved. In this paper, we study fcoloring on graphs ⊙ .

2. The Main Results

In this section, we show that the corona product of a complete graph with a cycle graph is in Cf 1, if the value of f is constant for all vertices. Before that, we give some definition. Let C be a color set. An edge colored with color c ∈ C is called an c-edge. The number of c-edges of G which incident with a vertex v is denoted by d(v,c). Let f(v) = k for every ∈ ( ), define m(v,c)= k – d(v,c). Color c is available at vertex v if m(v,c) ≥ 1. Theorem 3. Let G =

. For every n,m ∈ Ζ+ and f (v) = k for every v ∈ V(G) then G ∈ Cf 1.

Proof : Let C be a color set for f-coloring each edges E(G), where | C | = ∆f(G).


From the definition in eq. (4), we know that v ∈ V*, for every v ∈ K n . (1) For every v ∈ V \ V*, we have ( ) ( )

c = 1 ≤ Δ ( ), because V \ V* = V( K m ).

Hence, the pendant edges can be colored by colors in C. (2) Let v ∈ V* , we know that d(v) = n + m – 1. We divide the proof into two cases : Case 1. k Œ kd ( v ) , for every v. By theorem 1, then G ∈ Cf 1 Case 2: k d (v) , for every v. Let C* is a multiset, C* = {ci ∈ C | mulitiplicity of ci is k, for every i}. It’s clear that |C*| = k .α, where α = ∆f(G). First step, we assign colors in C* to all of edges which incident with v1 ∈ V*. The color c1 for k edges, color c2 for another k edges, and so on. Therefore, kα colors have been used to color the all of edges which incident with v1 ∈ V*. We obtained m(v, ci) = k – 1, for some i and for every v ∈ V* – {v1}. Second step, we assign colors in C* – {ci} to all of edges which incident with v2 ∈ V*. Now, m(v, ci) = k – 2, for some i and for every v ∈ V* – {v1, v2} . Generally, we can assign |C*| – (j –1) colors to all of edges which incident with vj ∈ V*, 1 ≤ j ≤ n. Thus, a desired f -coloring of G is obtained by ∆f colors.    Next theorem shows that the corona product of Kn with Cm to be of Cf 1, where f (v) = k for every v ∈ K n  Cm (m ≥ 3) . Theorem 4. Let n,m ∈ Ζ+. If


and f (v) = k for every v ∈ V(G), then G ∈ Cf 1.

Proof : Let C be a color set for f-coloring each edges E(G), where |C| = ∆f(G). * Case 1. k Œ kd ( v ) , for every v ∈ V By theorem 1, then G ∈ Cf 1 Case 2: k d ( v ) , for every v ∈ V* (a) When n = 1, we know that G = Wn, Then G ∈ Cf 1. (b) When n = 2, by theorem 2, Then G ∈ Cf 1. (c) When n > 2, let C* is a multiset, C* = {ci ∈ C | mulitiplicity of ci is k, for every i}. Assume, we erase all of edges of Cm in G. By using a similar method in the proof of Theorem 3, It is clear that we have colored the K n  K mc by ∆f colors. Finally, we assign colors for the paths. The edges of Cm are colored alternately with the colors which was used in E(Kn). Thus, a desired f -coloring of G is obtained by ∆f colors.   

3. Future Works In the future, we will give a sufficient condition for a graph the corona product of Kn with another graph. Therefore, we will work on f-coloring when f (v) is given for every vertex v ∈ V(G).

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