9 F eb 1 99 5 The Knowlton - Graham Partition Problem
نویسنده
چکیده
A set partition technique that is useful for identifying wires in cables can be recast in the language of 0–1 matrices, thereby resolving an open problem stated by R. L. Graham in Volume 1 of this journal. The proof involves a construction of 0–1 matrices having row and column sums without gaps. A long cable contains n indistinguishable wires. Two people, one at each end, want to label the wires consistently so that both ends of each wire receive the same label. An interesting way to achieve this was proposed by K. C. to the condition that at most one element appears both in an A set of cardinality j and in a B set of cardinality k, for each j and k. We can then use the coordinates (j, k) to identify each element. R. L. Graham [2] proved that such partitioning schemes exist if and only if n = 2, 5, or 9. By restating the problem in terms of 0–1 matrices, it is possible to prove Graham's theorem more simply, and to sharpen the results of [2]. Lemma 1. Knowlton-Graham partitions for n exist if and only if there is a matrix of 0s and 1s having row sums n} with the Knowlton-Graham property , let a jk be the number of elements that appear in an A set of cardinality j and a B set of cardinality k. Then a jk is 0 or 1; and r j = k a jk is j times the number of A sets of cardinality j, while c k = j a jk is k times the number of B sets of cardinality k. Conversely, given such a matrix, we can use its rows to define A 1 ,. .. , A p such that each 1 in row j is in an A set of cardinality j; similarly its columns define B 1 ,. .. , B q such that each 1 in column k is in a B set of cardinality k. has row and column sums (2, 6, 3, 4, 5, 6) that satisfy the divisibility condition and sum to 26. To identify 26 wires, we can associate the 1s with arbitrary labels
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تاریخ انتشار 1996