Path Coverings of Graphs and Height Characteristics of Matrices

Using graph theoretic techniques, it is shown that the height characteristic of a triangular matrix A majorizes the dual sequence of the sequence of differences of maximal cardinalities of singular k-paths in the graph G(A) of A and that in the generic case the height characteristic is equal to that dual sequence. The results on matrices are also used to prove a graph theoretic result on the duality of the sequence of differences of minimal k th norms of path coverings for a (0-1 )-weighted acyclic graph G and the sequence of differences of maximal cardinalities of k-paths in G. This result generalizes known results on unweighted graphs. © 1993 Academic