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Let G be a graph. A minimal coloring of G is a coloring which has the smallest possible sum among all proper colorings of G, using natural numbers. The vertex-strength of G, denoted by s(G), is the minimum number of colors which is necessary to obtain a minimal coloring. In this note we study these concepts, and define a new concept called the edge-strength of G, denoted by s′(G). We prove the celebrated Brooks' theorem for χ(G) replaced by s(G) and we also prove the following upper bound for s(G): s(G) ≤ ⌈col(G) + Δ(G)/2⌉, where col(G) is an invariant based on linear orderings of the vertices. Also, it is proved that s′(G) lies between Δ(G) and Δ(G) + 1, as for χ′(G), but it may be not... 

In this note a conjecture of P. Johnson Jr., on the Hall condition number is disproved  

In this study, a multi-item economic order quantity model with shortage under vendor managed inventory policy in a single vendor single buyer supply chain is developed. This model explicitly includes warehouse capacity and delivery constraints, bounds order quantity, and limits the number of pallets. Not only the demands are considered imprecise, but also resources such as available storage and total order quantity of all items can be vaguely defined in different ways. An ant colony optimization is employed to find a near-optimum solution of the fuzzy nonlinear integer-programming problem with the objective of minimizing the total cost of the supply chain. Since no benchmark is available in... 

The purpose of this paper is to develop a multi-item economic order quantity (EOQ) model with shortage for a single-buyer single-supplier supply chain under green vendor managed inventory (VMI) policy. This model explicitly includes the VMI contractual agreement between the vendor and the buyer such as warehouse capacity and delivery constraints, bounds for each order, and limits on the number of pallets. To create a kind of green supply chain, tax cost of green house gas (GHG) emissions and limitation on total emissions of all items are considered in the model. A hybrid genetic and imperialist competitive algorithm (HGA) is employed to find a near-optimum solution of a nonlinear... 

Recent researches have shown the importance of improving the supply chain competitiveness by means of strategic alliances. This study considers the retailer-supplier partnership through a vendor-managed inventory (VMI) system and develops an analytical model to explore the effect of important supply chain parameters on the cost savings realized from collaborative initiatives. This model is developed for a two-level supply chain consisting of a single supplier and a single retailer and examines the inventory management practices before and after implementation of VMI. The results of analytical examination show that VMI implementation in economic order quantity model when shortage is... 

This paper proves that if G is a graph (parallel edges allowed) of maximum degree 3, then χc′(G) ≤ 11/3 provided that G does not contain H1 or H2 as a subgraph, where H1 and H2 are obtained by subdividing one edge of K23 (the graph with three parallel edges between two vertices) and K4, respectively. As χc′(H1) = χ c′(H2) = 4, our result implies that there is no graph G with 11/3 < χc′(G) < 4. It also implies that if G is a 2-edge connected cubic graph, then χc′(G) ≤ 11/3. © 2005 Wiley Periodicals, inc  

At the 16th British Combinatorial Conference (1997), Cameron introduced a new concept called 2-simultaneous coloring. He used this concept to reformulate a conjecture of Keedwell (1994) on the existence of critical partial latin squares of a given type. Using computer programs, we have verified the truth of the above conjecture (the SE conjecture) for all graphs having less than 29 edges. In this paper we prove that SE conjecture is a consequence of the well-known oriented cycle double cover conjecture. This connection helps us to prove that the SE conjecture is true for semieulerian graphs. © 2000 Elsevier Science B.V. All rights reserved  

At the 16th British Combinatorial Conference (1997), Cameron introduced a new concept called 2-simultaneous edge-coloring and conjectured that every bipartite graphic sequence, with all degrees at least 2, has a 2-simultaneous edge-colorable realization. In fact, this conjecture is a reformulation of a conjecture of Keedwell (Graph Theory, Combinatorics, Algorithms and Applications, Proceedings of Third China-USA International Conference, Beijing, June 1-5, 1993, World Scientific Publ. Co., Singapore, 1994, pp. 111-124) on the existence of critical partial latin squares (CPLS) of a given type. In this paper, using some classical results about nowhere-zero 4-flows and oriented cycle double... 

Based on previous studies, empirical distribution of the bacterial burst size varies even in a population of isogenic bacteria. Since bacteriophage progenies increase linearly with time, it is the lysis time variation that results in the bacterial burst size variations. Here, the burst size variation is computationally modeled by considering the lysis time decisions as a game. Each player in the game is a bacteriophage that has initially infected and lysed its host bacterium. Also, the payoff of each burst size strategy is the average number of bacteria that are solely infected by the bacteriophage progenies after lysis. For calculating the payoffs, a new version of ball and bin model with...