A combinatorial problem involving graphs and matrices |
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Authors: | William T. Trotter Ted R. Monroe |
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Affiliation: | Department of Mathematics and Statistics, University of South Carolina, Columbia, SC 29208, USA |
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Abstract: | In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x1,x2,…,xk?nk there exist k distinct integers j1,j2,…,jk so that xi= aii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k). |
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