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Báo cáo toán học: "Sum List Coloring 2 × n Arrays"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài :Sum List Coloring 2 × n Arrays. | Sum List Coloring 2 X n Arrays Garth Isaak Department of Mathematics Lehigh University Bethlehem PA 18015 gisaak@lehigh.edu Submitted September 30 2001 Accepted May 7 2002. MR Subject Classifications 05C15 Abstract A graph is f -choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f. We show that the sum choice number of a 2 X n array equivalent to list edge coloring K2n and to list vertex coloring the cartesian product K2ũKn is n2 5n 3 . List coloring has been well-studied in recent years see for example surveys in 1 7 9 11 . The vertices of a graph often a line graph are given color lists and we seek to determine if there is a proper coloring using colors from the lists. Typically one seeks the minimum value k such that for every choice of lists of size k there is a proper coloring. In this case we call the graph k-choosable. More generally the list size for each vertex can be specified and one can determine if for every choice of lists of the specified sizes there is a proper coloring. The general problem here would be to characterize which list size assignments allow a proper coloring for every choice of lists of the given sizes. For the case of line graphs of complete bipartite graphs this is called the generalized Dinitz problem in 5 and it is noted that it was proposed independently by at least H. Taylor and D. Knuth personal communications and the author. The author referred to in the previous sentence is Galvin. The general characterization problem for line graphs of trees and more generally for vertex coloring graphs with every block a clique is solved in another paper 8 by the author of this paper. For the k-choosability problem we seek to minimize the maximum list size. Here we will examine another special case of the generalized problem where we seek to minimize the average list size