Let the star on n vertices, namely K1,n−1 be denoted by Sn. If every two coloring of the edges of a complete balanced multipartite graph Kj×s there is a copy of Sn in the first color or a copy of Sm in the second color, then we will say Kj×s → (Sn, Sm). The size Ramsey multipartite number mj(Sn, Sm) is the smallest natural number s such that Kj×s → (Sn, Sm). In this paper, we obtain the exact values of the size Ramsey numbers mj(Sn, Sm) for n, m 3 and j 3.

For simple graphs G 1 and G 2 , the size Ramsey multipartite number m j (G 1 , G 2 ) is defined as the smallest natural number s such that any arbitrary two coloring of the graph K j×s using the colors red and blue, contains a red G 1 or a blue G 2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers m j (nK 2 , C m ) for j ≥ 2 and m ∈ {3, 4, 5, 6}.

Let G and H be two simple subgraphs of s j K ×. The smallest positive integer s such that any red and blue colouring of s j K × has a copy of red G or a blue H is called the multipartite Ramsey number of G and H. It is denoted by) , (H G m j. This paper presents exact values for) , (2 G B m j where G is a isolate vertex free graph up to four vertices.

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It has been conjectured that r(Cm, Kn) = (m − 1)(n − 1) + 1 for all m ≥ n ≥ 4. This has been proved recently for n = 4 and n = 5. In this paper, we prove that r(C5, K6) = 21. This raises the possibility that r(Cm, K6) = 5m − 4 for all m ≥ 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 99–108, 2000

The government of Sri Lanka has established several economic centres in the provincesbased on the budget proposal which was released in the year 1998. The Dambulla economic centre was the first such centre which was established on the 1st of April 1999. Since then, a number of economic centres were established throughout the island. However, Dambulla main hub continues to be the central warehouse where the vegetables are stored in the island. This paper deals with vehicle scheduling related to transportation and it investigates a method whereby a solution can be arrived, using linear programming (LP). Marketing Department Logistics (MDL), Ltd. needs to distribute vegetables and fruits to different provinces. Its main hub is situated near the Dambulla vegetable and fruit market and minor hubs are situated in different provinces in Sri Lanka. The main objective of this research is building a cost minimized model which creates a suitable method for delivering vegetables and fruits from the Dambulla major hub through its minor hubs to outlets in the provinces. Hence, in order to optimize the cost of outbound distribution, a mathematical model has been developed by using Integer Linear Programmingand by using industrial based reliable sources to collect data. Software assistance was obtained using the LINGO 06 optimizer, Java, MS Access and MS Excel tools to solve this mathematical model. This study is based on the Dambulla economic centre. This is an initial step to bring a correct protocol to arrange a transport model in order to distribute the vegetables and fruits from this centre in a cost-effective manner. According to this study, all districts in Sri Lanka could be divided into four clusters. At the beginning of this research, we assumed that each district contains two warehouses and three vendors. This model paves the way to create a larger model for solving any type of transportation planning problem.

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Let P n represent the path of size n. Let K 1,m−1 represent a star of size m and be denoted by S m . Given a two coloring of the edges of a complete graph K j×s we say that K j×s → (P n , S m+1 ) if there is a copy of P n in the first color or a copy of S m+1 in the second color. The size Ramsey multipartite number m j (P n , S m+1 ) is the smallest natural number s such that K j×s → (P n , S m+1 ).Given j, n, m if s = n + m − 1 − k j − 1 , in this paper, we show that the size Ramsey numbers m j (P n , S m+1 ) is bounded above by s for k = n − 1 j . Given j ≥ 3 and s, we will obtain an infinite class (n, m) that achieves this upper bound s. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.