• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.49-58
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• 2009

For $1{\leq}k{\leq}r$, let ${\sigma}$($K_{r,r}$ - ke, n) be the smallest even integer such that every n-term graphic sequence ${\pi}$ = ($d_1$, $d_2$, ..., $d_n$) with term sum ${\sigma}({\pi})$ = $d_1$ + $d_2$ + ${\cdots}$ + $d_n\;{\geq}\;{\sigma}$($K_{r,r}$ - ke, n) has a realization G containing $K_{r,r}$ - ke as a subgraph, where $K_{r,r}$ - ke is the graph obtained from the $r\;{\times}\;r$ complete bipartite graph $K_{r,r}$ by deleting k edges which form a matching. In this paper, we determine ${\sigma}$($K_{r,r}$ - ke, n) for even $r\;({\geq}4)$ and $n{\geq}7r^2+{\frac{1}{2}}r-22$ and for odd r (${\geq}5$) and $n{\geq}7r^2+9r-26$.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.119-126
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• 1996

In this paper we study the asymptotic behavior of the edge independence number of a random (n,n)-tree. The tools we use include the matrix-tree theorem, the probabilistic method and Hall's theorem. We begin with some definitions. An (n,n)_tree T is a connected, acyclic, bipartite graph with n light and n dark vertices (see [Pa92]). A subset M of edges of a graph is called independent(or matching) if no two edges of M are adfacent. A subset S of vertices of a graph is called independent if no two vertices of S are adjacent. The edge independence number of a graph T is the number $\beta_1(T)$ of edges in any largest independent subset of edges of T. Let $\Gamma(n,n)$ denote the set of all (n,n)-tree with n light vertices labeled 1, $\ldots$, n and n dark vertices labeled 1, $\ldots$, n. We give $\Gamma(n,n)$ the uniform probability distribution. Our aim in this paper is to find bounds on $\beta_1$(T) for a random (n,n)-tree T is $\Gamma(n,n)$.

• Journal of Korean Institute of Industrial Engineers
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• v.24 no.4
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• pp.649-660
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• 1998

Steel making in Mini Mill consists of three major processing stages: molten steel making in an electric arc fuenace, slab casting in a continuous caster, and hot rolling in a finishing mill. Each processing stage has its own lot grouping criterion. However, these criteria in three stages are conflicting with each other. Therefore, delveloping on efficient lot grouping algorithm to enhance the overall productivity of the Mini Mill is an extremely difficult task. The algorithm proposed in this paper is divided into three steps hierarchically: change grouping, cast grouping, and roll grouping. An efficient charge grouping heuristic is developed by exploiting the characteristics of the orders, the processing constraints and the requirements for the downstream stages. In order to maximaize the productivity of the continuous casters, each cast must contain as many charges as possible. Based on the constraint satisfaction problem technique, an efficient cast grouping heuristic is developed. Each roll consists of two casts satisfying the constraints for rolling. The roll grouping problem is formulated as a weighted non-bipartite matching problem, and an optimal roll grouping algorithm is developed. The proposed algorithm is programmed with C language and tested on a SUN Workstation with real data obtained from the H steel works. Through the computational experiment, the algorithm is verified to yield quite satisfactory solutions within a few minutes.