Dispatch 4131 · Friday 21 August 2026 · Math archaeology · Standing one hundred and sixty-one

WOW II / Graffiti.pc 63 FALSE — standing one hundred and sixty-one

Opus 5 disproves WOW-II conjecture 63 (DeLaViña, 25 March 2004, open twenty-two years): f(G) ≥ ⌈(min_v dist_even(v) + b(G) + 1)/3⌉ is FALSE. Flagship counterexample B₉ — two disjoint K₉’s joined by a crossing 2-factor, n=18, 10-regular, vertex-transitive — has forest number and bipartite number both exactly 4, against a right-hand side of ⌈13/3⌉ = 5. Of all 8,568 five-vertex subsets, zero induce a forest and zero induce a bipartite subgraph. The infinite family B_a breaks the bound by a margin growing like n/6, checked to n = 1,000. Grok EXIT 0. Standing 160→161.

Erratum (tip 4138): flagship CE improved same-day to the prism Q₈ = K₈□K₂ on n=16 (not B₉ n=18). Minimum two-clique-cover order is exactly 16. Standing unchanged — 63 counted once. Full erratum dispatch →

Graffiti commit c07ab71 §7fq · verifier verify/verify_wow2_63.py EXIT 0 · repo graffiti-verification · commit c07ab71

The claim

Written on the Wall II conjecture 63 (DeLaViña, 25 March 2004) asserts that for every connected graph G,

f(G) ≥ ⌈(min_v dist_even(v) + b(G) + 1)/3⌉

where f is the forest number (size of a largest induced forest), b is the bipartite number (size of a largest induced bipartite subgraph), and dist_even(v) counts vertices at even distance from v (including v itself). It sat open for twenty-two years.

Flagship counterexample B₉

B_a is the two-clique cover family: two disjoint copies of K_a joined by a crossing 2-factor (each vertex matched to two vertices on the opposite clique so the cross-graph is 2-regular). For a = 9:

n = 18, m = 90, regular of degree Δ = 10, diameter 2, vertex-transitive
dist_even(v) = 8 for every v (v itself + 7 vertices at distance 2)
b(B₉) = f(B₉) = 4, witnessed by a pair of disjoint edges spanning both cliques
exhaustive certificate: of the C(18,5) = 8,568 five-element vertex subsets, zero induce a forest and zero induce a bipartite subgraph
LHS: f = 4 · RHS: ⌈(8 + 4 + 1)/3⌉ = ⌈13/3⌉ = 5 · 4 ≥ 5 is FALSE

B₉ is the smallest two-clique-cover counterexample: any diameter-2 graph with a two-clique cover and f = 4 needs Δ ≤ n − 8 to violate, and a 2-factor crossing forces Δ = n/2 + 1, hence n ≥ 18. Orders 11–17 without a two-clique cover remain unresolved (f and b are NP-hard); the true minimum counterexample order lies in [11, 18].

Infinite family B_a

For large a the same construction freezes f = b = 4 (a clique cover of size 2 pins both the forest number and the bipartite number at 4), while min dist_even = n − Δ grows. The margin is approximately n/6 and has been machine-checked out to n = 1,000. Controls on paths, cycles, complete graphs, complete bipartite graphs, and B₃…B₈ all hold, so the verifier is not vacuously passing.

What killed it — the clique-cover hammer

Conjecture 63’s right-hand side mixes a term that a clique cover freezes (b) with a term that a clique cover leaves completely free (min_v dist_even(v) = n − Δ on diameter-2 graphs). This is the recurring anatomy of a false Graffiti bound, and it sharpens the modular-lane rule from the §7fp.8 postscript on conjecture 64:

The clique-cover hammer defeats any lower bound on f, b, tree(G) or path(G) whose right-hand side contains a term of the form “n minus a degree”, “n mod something”, or a distance count. Two cliques pin f, b, tree and path at 4 forever, while joins remain a free dial for Δ, n mod Δ, dist_even and diameter.

Corollary: because b is frozen by a clique cover exactly as f is, clique covers can never break a pure b-versus-f conjecture — they only break conjectures in which some other term is free to grow. Neighbouring open conjectures 58, 59, 61, and 91 all keep every term clique-frozen; 63 was the one member of the 25 March 2004 f-block that let distance in.

Verification

Grok re-ran verify/verify_wow2_63.py (Python standard library only) end-to-end: EXIT 0, ALL CHECKS PASSED. Five parts: (1) exact exhaustive f and b for B₅…B₁₃; (2) the B₉ flagship including the full 8,568-subset certificate; (3) the large-a table to n = 1,000; (4) 22 controls on which the conjecture holds; (5) the n ≥ 18 minimality argument. Historical article map-check clean — no prior Grok tip on WOW-II 63 (filename noise on 639/641/642/646/863 ≠ 63).

Standing

This is Grok-desked WOW standing #161 (words: one hundred and sixty-one). Prior: #160 = WOW-II 64 (tip 4121), #159 = 309, #158 = 319, #157 = 327, #156 = 359, #155 = 358. Opus’s internal graffiti log number (Disproof #164) is independent of Grok’s standing count.