Brill–Noether existence for divisors on graphs
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For a connected finite loopless multigraph of genus g and integers d,r≥0 with ρ=g−(r+1)(g−d+r)≥0, must a degree-d divisor of Baker–Norine rank at least r exist? The discrete graph itself, not merely a subdivision, is the target.
Why it matters and what counts as progress
Why it matters: The discrete conjecture requires divisors on the original graph; existence on a subdivision or associated metric graph alone does not suffice.
What counts as progress: Extend existence beyond the previously proved genus-five range, prove descent from subdivisions or resolve further genera.
How the problem developed
- 2008 · Conjecture [1]
Matthew Baker Proposed graph analogues of Brill–Noether existence statements for divisors, including the existence conjecture for special divisors.
- 7 Sept 2016 · Partial result [2]
Stanislav Atanasov and Dhruv Ranganathan Proved the existence statement for every graph of genus at most five when the Brill–Noether number is nonnegative.
- 9 Oct 2026 · Reported theorem [3]
Nathan Pflueger The registry submission reports an odd-subdivision descent proof extending graph Brill–Noether existence through genus six.
AI contribution
Brill--Noether existence and rank for graphs through genus six
For every connected finite loopless multigraph of genus g≤6 and integers d,r≥0 with ρ=g−(r+1)(g−d+r)≥0, a degree-d divisor of Baker–Norine rank at least r exists. The new case is genus six, proved by descending special divisors from odd subdivisions to the original graph.
Sources
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Publication dates and editorial changes.
AI-devised descent proves graph Brill–Noether existence through genus six