Bipartiteness criterion A graph is bipartite if and only if it does not contain odd cycles. Is there a similar criterion for hypergraphs? (A hypergraph is called bipartite if its vertices can be colored in two colors so that no hyperedge is monochromatic.)
My guess is that the answer is "no" but, maybe, there are results in this direction I am not aware of. Thanks!
 A: No, there isn't because deciding whether a hypergraph is bipartite (2-colorable) or not is NP-complete (reduction from NAE-SAT) already for 3-uniform hypergraphs. Of course if by similar you mean something that is not necessarily verifiable in polynomial time, then the answer might be yes...
A: If one instead starts with the definition of bipartite as not containing any odd cycles, then there are results for hypergraphs in this direction.  Indeed we can define a cycle in a hypergraph as a sequence of distinct alternating vertices and hyperedges $C:=v_1, E_1, v_2, E_2, \dots, v_n, E_n, v_1, E_1$, where $v_i \in E_i \cap E_{i+1}$ for all $i$ (mod $n$). The length of $C$ is $n$.  The following is a theorem of Gyarfas, Jacobson, Kezdy and Lehel.
Theorem. Let $\mathcal{H}$ be a hypergraph where each hyperedge contains at least 3 vertices.  If $\mathcal{H}$ contains no odd length cycles, then 
$$
\sum_{e \in E(\mathcal{H})} (|e|-1) \leq 2|V(\mathcal{H})| -2.
$$
Moreover, equality holds if and only if $\mathcal{H}$ is the disjoint union of two identical uniform hypertrees.  
A: In "matching theory", by Lovasz and Plummer, I found the following page, which may be helpful:

