Zero-Sum Quartet Incidence and Sharp Maximal-Independent-Set Witnesses on Eight Vertices
We prove an exact four-term probability law for zero-sum maximal-independent-set witnesses. For every connected simple graph G on eight vertices and every admissible eight-point cap U in F_3^3, sample uniformly from all 40,320 bijections of the vertices onto U. Let N0 count zero-sum maximal independent four-sets, m count the size-four maximal independent sets, n0 and n2 count their unordered pairs intersecting in zero and two vertices, and h count compatible unordered triples. Put M1 = 2m/35, M2 = 2n0/35 + n2/315, and M3 = h/315. Then P(N0=0) = 1-M1+M2-M3, P(N0=1) = M1-2M2+3M3, P(N0=2) =…
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