arXiv subjects

Haroldo Costa Silva Filho

Publications and source records attributed to Haroldo Costa Silva Filho.

3 recordsLinked to original sources

arXiv2026

The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields

The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two problems into one. Where does it live? As a Minkowski sum of unit triangles, each H(d,3) has a hidden flexibility carrying its coordinates up the constructibility ladder (compass, origami, and beyond), so one graph is realizable over many number fields at once, in the plane and in R^3 (edim(H(d,q))=q-1). Yet almost every faithful realization is transcendental: the Galois picture is a measure-zero shadow of a vast transcendental continuum, matching the exists-R hardness of unit-distance recognition. How crowded must a unit drawing be? We separate the ordinary crossing number from a unit-distance crossing number (absent from Schaefer's survey), and one recursion H(d,3)=H(d-1,3) box K_3 controls both. We prove cr(H)=Theta(n^2), with sharp one-page constant 7/6; and Omega(n^2) <= udcr(H) <= O(n^2 log n) by concentration, plus a closed-form majorant (3/2)n^2(L^2-L+1), L=log_3 n. The least-crossing drawing we find is constructible (origami): field and crossings are two readings of one geometry. All claims are verified computationally; the lower bound udcr=Omega(n^2 log n) is the central open problem.

math.CO↗

arXiv2026

Origametry and antagonistic heptagon graphs: the braced {35,1} and the complete {21,2}

We study two planar unit-distance graphs (UDGs) that both look like heptagons but have opposite rigidity yet kindred arithmetic: the braced heptagon {35,1} (19 vertices, 35 edges) and the complete heptagon {21,2} (21 vertices, 42 edges). Neither is straightedge-and-compass constructible -- already the regular 7-gon is not, by Gauss-Wantzel -- so the natural setting is origametry: the construction of each realization by Huzita-Hatori folds, with the quadratic fold O5 (straightedge-and-compass strength) versus the cubic Beloch fold O6. The complete heptagon is over-braced and globally rigid, with a unique realization whose 42 edges are certified unit exactly over Q(cos 2pi/7), a cyclic cubic field: every vertex is an origami number, reached by one Beloch fold. The braced heptagon is isostatic (minimally rigid) with thousands of non-congruent realizations and a large generic realization count N = 3869504 = 2^6 * 103 * 587; yet we prove that its unit realization is again origami-constructible. In Pegg's explicit construction the pinning angle is a root of an irreducible degree-12 palindromic polynomial whose Chebyshev reduction, an irreducible sextic, factors into quadratics over the heptagon cubic Q(cos 2pi/7); hence the coordinate field is a {2,3}-tower of degree 3*2^k, reached by one trisection followed by quadratics. The primes 103 and 587 therefore belong to the generic complex count, not to the folded coordinates. We solve the governing cubic explicitly by folding (Lill's method and the Beloch fold), place both graphs inside Alperin's field of origami numbers, and read the fold-branch signs of the construction as mountain-valley assignments with an empirical Maekawa-type balance. All claims are checked in exact arithmetic.

math.MG↗

arXiv2026

Exact certification of the coordinate fields of the triangle-free Exoo-Ismailescu unit-distance graphs EI17 and EI19 (HoG 51375, 51376): a solvable-non-solvable dichotomy (origami vs. S20) and the Laman-number conjecture

We certify, exactly, the coordinate fields of a faithful planar realization of two neighbouring triangle-free Exoo-Ismailescu unit-distance graphs (UDGs), and show they realize the two opposite extremes of the constructibility hierarchy. The 17-vertex graph EI17 (House of Graphs 51375) is the smallest triangle-free UDG with chromatic number 4; the 19-vertex graph EI19 (HoG 51376) is its state-of-the-art origami neighbour. In both, fixing a rational base edge, the remaining vertices are intersections of unit circles -- each on the radical axis of its two neighbours, a tower of square roots over the free angles -- and a small closure system locks the realization. For EI19 the base lies in Q(sqrt 2, sqrt 5, sqrt 7) and a single free angle has an irreducible degree 12 = 2^2*3 minimal polynomial with Galois group the solvable transitive group 12T236 (order 2304 = 2^8*3^2): not ruler-and-compass, but origami-constructible (the cubic Beloch fold O6 necessary, in casus irreducibilis). For EI17 two free angles are locked by two closures, whose resultant is irreducible of degree 20 = 2^2*5 with Galois group the full symmetric group S20 (a Frobenius census exhibits a 17-cycle, forcing A20 by Jordan, and an odd 20-cycle, raising it to S20): non-solvable, so the coordinates are not expressible by radicals -- neither compass nor origami of any fold order. Thus the smallest triangle-free 4-chromatic UDG is the generic, maximally exotic case, the exact opposite of its origami neighbour. We give the full certification pipeline as explicit algorithms, record two methodological pitfalls, and read the pair through a conjectural bridge between the Laman number and the Galois group.

math.CO↗