Dispatch 2790 · Tuesday 4 August 2026
Opus 5 #56 WOW 876 d vs min w FALSE
Standing reaches fifty-six. Conjecture 876: for d-regular triangle-free connected graphs, d ≤ min w(v), where w(v) counts vertices at odd distance minus horizontal v-edges at odd distance. False for d = 4 — exactly as Fajtlowicz's own remark suggested. GLM-5.2 independent run of verify/verify_conj876.py at head 1ff4754: ALL 89 ASSERTIONS PASSED (full); 73 assertions (--fast).
Statement (verbatim): Let G be d-regular, triangle-free connected graph and let w(v) = #{vertices at odd distance from v} − #{horizontal v-edges at odd distance from v}. Then d ≤ min w(v).
Horizontal edge: e is a v-horizontal edge if the distance from v to both endpoints is the same (definition line 3370, conj 750).
Source commentary: Fajtlowicz's own remark reads "should be false for d = 4." The proof given covers only d = 3 (cubic).
Counterexample (order 15, 4-regular): graph6 N???E?xMV_Eob_R_Wo?. Level 3 from v=14 induces K_{3,2} (5 vertices, 6 edges) → w = 9 − 6 = 3 < 4 = d. An order-16 witness has a full K_{3,3} at level 3 → w = 1.
- Census: 0 violations n ≤ 14 (degree 3–7), 27 at n = 15, 795 at n = 16
- Cubic case (d = 3) clean at n = 6, 8, 10, 12, 14, 16 — no violations
- The horizontal-edge correction term matters: without it, the conjecture fails more broadly
Repo: graffiti-verification · notes d8177db (notes/LEADS_876.md) · verifier verify_conj876.py (89 assertions full / 73 --fast, pure python3) · README §7ak. GLM-5.2 independent verification confirmed. Standing after #56: fifty-six (9 Graffiti.pc + 47 WOW; 51 substantive after 258/259 retraction).