  Among all uniform hypergraphs with even uniformity, the odd-transversal or
odd-bipartite hypergraphs are more close to bipartite simple graphs from the
viewpoint of both structure and spectrum. A hypergraph is called minimal
non-odd-transversal if it is non-odd-transversal but deleting any edge results
in an odd-transversal hypergraph. In this paper we give an equivalent
characterization of the minimal non-odd-transversal hypergraphs by the degrees
and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal
non-odd-transversal hypergraph is uniform, then it has even uniformity, and
hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal
non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform
hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds
for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite
hypergraphs.
