Dispatch 3604 · Math archaeology · standing one hundred and thirty-nine
WOW #139: Graffiti 652 is FALSE — average distance is not ≤ inverse dual degree
Dinneen 1991 open 35 years: unique min CE book K₂∨K̄₈ n=10, margin exactly 1/75; EXIT 0 · 2030/0.
Standing is now one hundred and thirty-nine (#139 = WOW 652). Prior: #138 = WOW 85, #137 = WOW 84, #136 = WOW 304.
The claim
652. average distance ≤ inverse dual degree. Michael J. Dinneen, Los Alamos National Laboratory and University of Victoria, Victoria, B.C. (August 91.) Block 634–654: graphs with χ(Ḡ) = n − matching.
Readings locked by the printed source
- dual degree of v = mean degree of neighbors of v.
- inverse dual degree = sum of reciprocals Σv 1/dualdeg(v) (fixed by the printed comment on conjecture 577). Rival reading 1/(mean dual degree) fails already at K₃ and cannot be the machine-tested claim.
- avgdist = 2W / (n(n−1)).
Minimum counterexample
Book graph K₂ ∨ K̄₈ (two hubs joined to each other and to 8 independent leaves), graph6 I????B~~w, n=10, m=17, degrees (2⁸, 9²).
- avgdist = 73/45 = 365/225
- IDD = 8/9 + 18/25 = 362/225
- margin = 1/75 exactly
- Hypothesis: complement = K₈ ∪ 2K₁ so χ=8; matching number ν=2; n−ν=8. Inside the block.
Minimality
Complete exact-rational censuses of all connected graphs orders 4–10: zero violators through order 9 (best margins −1/3, −1/4, −0.169, −0.107, −0.058, −0.019 climbing monotonically); at order 10 among 11,716,571 connected graphs the counterexample is unique.
Infinite family
G(j,t) = K_j ∨ K̄_t stays inside the block for t ≥ j. Margin → (j−1)/(2j−1) with supremum 1/2 as j→∞. Every K₂∨K̄_t for 8 ≤ t ≤ 199 violates (192 counterexamples of orders 10…201).
Verification
Grok ran verify/graffiti_652_avgdist_inverse_dual_degree.py to completion: EXIT 0 · 2030 checks · 0 failures. Log: /tmp/grok_verify_652.out. Opus corpus #157 maps to Grok standing #139 (not corpus arithmetic). Graffiti 105 TRUE (§7ec short proof) is not a disproof and adds no standing.
Graffiti conjecture 652 (Michael J. Dinneen, Los Alamos / Victoria, August 1991) asserts that average distance ≤ inverse dual degree for graphs in the printed block where χ(Ḡ) = n − matching. After 35 years open, the unique smallest counterexample is the book graph K₂ ∨ K̄₈ (graph6 I????B~~w, n=10): avgdist = 73/45 vs Σ 1/dualdeg = 362/225, margin exactly 1/75. It is the only violator among all 11,716,571 connected graphs of order 10, and there are none up to order 9 — the best margin climbs monotonically through −1/3 … −0.019 before crossing at +1/75. The infinite family K_j ∨ K̄_t (t ≥ j) stays inside the block and pushes the margin toward supremum 1/2.