The incidence matrix of a graph $G = (V,E)$ is a matrix with $|V|$ rows and $|E|$ columns, in which element $v,e$ is $1$ if node $v$ is incident to edge $e$, and $0$ otherwise.
In bipartite graphs, the incidence matrix has the following property (called "total unimodularity"): the determinant of every square submatrix is either $-1$ or $0$ or $1$. This property is very useful, because it implies that every linear program in which the constraints are based on this matrix (for example: for finding a maximum cardinality matching, maximum weight matching or minimum vertex cover) has a solution in which all values are integers.
I am now researching tripartite hypergraphs. Their incidence matrix need not be totally unimodular. For example, consider the hypergraph on the vertices $\{1,4,2,5,3,6\}$ with hyperedges: $\{ \{4,5,3\}, \{4,2,6\}, \{1,5,6\} \}$. Its incidence matrix is 6 by 3:
\begin{pmatrix}0& 0& 1\\0 &1 &0\\1& 0& 0\\1& 1& 0\\1& 0& 1\\0& 1& 1\end{pmatrix}
The bottom submatrix is a 3-by-3 square matrix whose determinant is $-2$.
Is there another property, weaker than total unimodularity, which characterizes the incidence matrices of tripartite hypergraphs?
Cross-posted from math.SE
