König's theorem and bimatroids

Abstract

König's theorem asserts that the minimal number of lines (i.e., rows or columns) which contain all the ones in a 0-1 matrix equals the maximal number of ones in the matrix no two of which are on the same line. The theorem occupies a central place in the theory of matchings in graphs. An extension of König's theorem to "mixed matrices" has recently been given by Murota, and it generalizes a determinantal version of the Frobenius-König's theorem obtained earlier by Hartfiel and Loewy. These results are generalized. We consider the setup in which there are two finite sets X and Y and a bimatroid (or linking system) defined on the pair (X, Y). We then prove a minimax theorem for the rank function of the bimatroid which includes some earlier extensions of König's theorem.