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Is the zero forcing number of every connected graph with maximum degree at most its independence number plus one? A connected 24-vertex subcubic graph with independence number and zero forcing number refutes this 2017 TxGraffiti conjecture, and a 36-vertex cubic variant refutes the cubic form: is attained.
Infinite family of counterexamples; mathematical argument internally checked, with external verification and novelty review pending.
For an odd prime , do Sun's normalized trigonometric permanents satisfy and ? Exact computation at refutes both sign laws.
Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support and conjectured that it was not a complete classification because it did not use the full field . This work proves the complete classification. The larger field enlarges the common amplitude , while every relative ambiguity remains a norm-one parameter in . Th…
Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph perfectly divisible if and only if for every induced subgraph of ? False: the Paley graph satisfies the chromatic bound hereditarily but is not perfectly divisible.
no constant-bound repair of the conjecture is possible
Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.
Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: "The proof turns conjugation, near , into base- arithmetic." A cycle near the fixed point splits into a coarse base permutation and a vector of carries in ,…
A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different. Question: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size? Answer: Ex…