Combinatorics and Graph Theory

    

Chordal Bipartite Graphs Are Rank Determined

Authors: Yaroslav Shitov

A partial matrix A is a rectangular array with entries in F ∪ {∗}, where F is the ground field, and ∗ is a placeholder symbol for entries which are not specified. The minimum rank mr(A) is the smallest value of the ranks of all matrices obtained from A by replacing the ∗ symbols with arbitrary elements in F. For any bipartite graph G with vertices (U, V), one defines the set M(G) of partial matrices in which the row indexes are in U, the column indexes are in V, and the (u, v) entry is specified if and only if u, v are adjacent in G. We prove that, if G is chordal bipartite, then the minimum rank of any matrix in M(G) is determined by the ranks of its fully specified submatrices. This result was conjectured by Cohen, Johnson, Rodman, Woerdeman in 1989.

Comments: 16 Pages.

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[v1] 2021-03-16 11:07:48

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