Validity of the Partition-Testing Conjecture for Bigraphic Degree Sequences

Prove or refute the conjecture that, for a degree sequence and a selected small set of balanced partitions tested by the Gale–Ryser characterization, failure of all those tests implies that no partition of the sequence is bigraphic.

Background

Recent approaches to BDR attempt to identify a small collection of balanced partitions whose Gale–Ryser tests could decide whether the entire degree sequence is bigraphic. The paper states that when every partition in this collection fails the test, it is conjectured that no other partition is bigraphic.

The conjecture is known in the special case where a partition perfectly separates the degrees into small and large values, but the general claim is not established in the paper and remains an unresolved question about whether a small tested set can certify non-big raphicity.