Papers

We prove that every three-dimensional parallelohedron $P$ satisfies \[ \mathcal H^2(\partial P)\ge 12\sqrt2\,r(P)^2, \] where $r(P)$ is the inradius of $P$, with equality if and only if $P$ is similar to the regular rhombic dodecahedron.

Given a finite preorder $\tau$, we study how its order relation determines the $h^*$-polynomial of the associated polar preorder polytope. Let $\mathcal R_{\tau}^{\vee}$ be the convex hull of the negative standard basis vectors and the indicator vectors of nonempty order ideals, and write $h_\tau^*(t)=h^*(\mathcal R_{\tau}^{\vee},t)$. Athanasiadis and Chapoton conjectured that this enumerative invariant takes ordinal sums to products: $$h_{\tau_1\oplus\tau_2}^*(t)=h_{\tau_1}^*(t)\,h_{\tau_2}^*(t).$$ We prove the conjecture for arbitrary finite preorders, including those with nontrivial equivalence classes. More precisely, we construct a positively homogeneous piecewise unimodular shear from the free sum $\mathcal R_{\tau_1}^{\vee}\oplus\mathcal R_{\tau_2}^{\vee}$ onto $\mathcal R_{\tau_1\oplus\tau_2}^{\vee}$, giving lattice-point bijections at every nonnegative real dilation scale. The key is a formula for the Minkowski functional in terms of the least nonnegative order-reversing majorant: after the shear, the functional becomes the sum of the two factor functionals. Together with the Ehrhart product formula for free sums, this proves the conjecture and the corresponding product formula for normalized volume.

We disprove a weighted generalization of the union-set conjecture [Fra95] raised in [Gow16]. We construct nonnegative weights on subsets of a finite ground set satisfying $w(\varnothing)=0$ and $w(A\cup B)\ge\min\{w(A),w(B)\}$, while every coordinate belongs to an arbitrarily small fraction of the total weight.

We introduce Latin Eulerian numbers $\LE{n\atop k_1,\dots,k_n}$, a multivariate refinement of classical Eulerian numbers counting order-$n$ Latin squares by column ascents. We establish their fundamental symmetries, univariate marginals, and an exact multiplicity divisibility property. For the total ascent statistic $\Sigma(L)=\sum_i k_i(L)$, we prove the sharp, isolated bounds $n-1 \le \Sigma(L) \le (n-1)^2$, demonstrating that the adjacent values $n$ and $(n-1)^2-1$ are strictly unattainable. To analyze intermediate values, we show that symbol permutations typically ignored in classical enumeration directly govern $\Sigma$ via an exact formula. This insight yields explicit constructions for the interior range, motivates a unimodality conjecture for the total ascent distribution.

We use stationary common-neighborhood properties to study highly connected Ramsey relations and partition hypotheses. For weakly compact $\kappa$, $\operatorname{Coll}(\omega_1,{<}\kappa)$ forces $\omega_2\to_{\mathrm{hc},<5}(\omega_2)^2_\omega$ and $\operatorname{PH}_1(\omega_2)$. If $\kappa$ is $T^{\kappa^+}_{\omega_1}$-Ramsey, the same collapse forces that every countable coloring of $[\omega_2]^2$ has a stationary set $X\subseteq\omega_2$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. From one weakly compact cardinal, we obtain a model of the ${<}5$-edge relation at $\omega_3$ and $\operatorname{PH}_1(\omega_3)$, in which $\check H^2(\omega_3,A_d)\ne0$ for every nontrivial abelian group $A$. This separates $\operatorname{PH}_1(\omega_3)$ from $\operatorname{PH}_2(\omega_3)$, with the exact consistency strength of one weakly compact cardinal. We also show that $\operatorname{PH}_1(\omega_2\times\omega_5)$ is equiconsistent with two weakly compact cardinals.

We reconstruct the generalized hexagon of order $(2,8)$ from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of $F_4(\mathbb R)$ and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of $N(A_1^{24})$ with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which $L_3(2)\cong L_2(7)$ acts as on $\mathbf P^1(\mathbf{F}_7)$; two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ${}^3D_4(2):3$, with a central factor $C_2$ for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.

This paper studies symmetry-induced reductions of scalar recurrence complexity for balanced banded Toeplitz determinants and permanents. Three mechanisms emerge: determinant state and spectral compression, exceptional permanent–determinant conversion, and symmetries of permanent state spaces. For symmetric determinants, straightening reduces the row-column states from $\binom{2m}{m}$ to the Catalan number $C_{m+1}$; primitive symplectic weight compression then leaves $(3^m+1)/2$ distinct autonomous modes. Skew symmetry has a parallel compound/Hodge explanation: the middle exterior representation splits into two Hodge halves of dimension $\binom{2m}{m}/2$, the one-step compound transfer exchanges the halves, and its two-step restriction has $3^{m-1}$ generic ternary modes. Thus the full skew bound is $2\cdot3^{m-1}$. Widom–Hankel arguments prove generic scalar minimality in both symmetry classes, and every even skew Toeplitz determinant admits an explicit half-size square factorization. For the zero-diagonal pentadiagonal support, a Pólya–Kasteleyn signing converts every permanent to a determinant. Among consecutive zero-diagonal two-sided bands, universal entrywise conversion—and separately Toeplitz diagonal-wise conversion—occurs only in the Hessenberg families and this pentadiagonal case. Paired renewal identities recover all restored-diagonal determinant and permanent layers. For permanents, transposition gives the open symmetric bound $(\binom{2m}{m}+2^m)/2$, while cyclic defect-sector pairing gives $(4^m+\binom{2m}{m})/2$. Skew-symmetry forces odd-size vanishing and corresponding even-subsequence bounds. In semibandwidth two the open symmetric bound is generically sharp; higher-semibandwidth permanent minimality is left separate from the symmetry reductions proved here.

We prove that the lattice covering density of the regular octahedron is $9/8$, settling a conjecture of Dougherty and Faber.

An arrangement of $n$ lines in the plane defines cells, i.e., connected components of the complement of the arrangement. Answering a question of Volkmar Welker we show that, apart from two special cases, there is always a line $L$ in the arrangement such that one of the open halfplanes whose boundary is $L$ contains more than half of the bounded cells of the arrangement. We also consider some unusual discrepancy problems that are related to this question.

For $k, r, n$ natural numbers let $\rho_{r,k}(K_n)$ be the number of $r$-edge-colorings of $K_n$ that do not contain a rainbow copy of a $K_k$, that is, a copy of $K_k$ in which all edges receive different colors. When $k=3$, the quantity $\rho_{r,3}(K_n)$ represents the number of Gallai Colorings. It was proved by Balogh and Li and independently by Bastos, Benevides and Han, that most of the Gallai colorings are 2-colorings, for $n$ large. A natural analogue conjecture would be that when $k=4$, $r\ge 5$ and $n$ large, most rainbow-$K_4$-free $r$-edge-colorings are $5$-colorings. We show that this is not true in general and identify an exact threshold for $r$ where this ceases to be true. For the range where the conjecture is false, we determine the exponential growth of $\rho_{r,4}(K_n)$ for every fixed $r$. More precisely, for \(6\le r\le24\), we prove that \(\rho_{r,4}(K_n)=(\binom{r}{5}+o(1))5^{\binom{n}{2}}\); and for each \(r\ge25\), the proportion using at most five colors tends to zero, and \(\rho_{r,4}(K_n)=r^{(n^2/4)+o(n^2)}\). A bipartite construction, with all edges within the two parts assigned one common color, achieves the latter exponential growth rate. The lower bounds can be easily generalized for every $k$. Those results are related to other recent results about counting colorings that avoid rainbow cliques or given rainbow patterns in general. Our proof combines hypergraph containers with the graph removal lemma, structural estimates for color palettes and a refined count of colorings close to a fixed five-color palette.

In his pioneering work from 1973, Delsarte conjectured that the Johnson scheme J$(n,w)$ admits no nontrivial perfect codes. Although related existence problems have been resolved in several other association schemes, the problem in the Johnson scheme remained open. We prove Delsarte's conjecture: no nontrivial e-perfect code exists in any Johnson scheme. The proof combines divisibility properties of balls in the scheme, bounds on the strength of designs associated with perfect codes, the roots of Lloyd polynomials, and explicit results on primes and linear forms in logarithms.

Let $t \ge 1$ be an integer, and let $G$ be a $t$-tough $n$-vertex graph with degree sequence $d_1, d_2, \ldots, d_n$ in non-decreasing order. In 1995, Hoàng conjectured that if $G$ is Hamiltonian and, for every integer $i$ satisfying $t\le i<n/2$, $d_i\le i$, and $d_{n-i+t}<n-i$, one has $d_j + d_{n-j+t} \ge n$ for all $j$ with $i < j < \frac{n}{2}$, then $G$ is pancyclic or bipartite. In this paper, we disprove the conjecture for $t = 1$ and confirm it for all $t \ge 7$.

Data stored in synthetic DNA is retrieved by shotgun sequencing, which returns short subsequences rather than the stored word itself. A natural abstraction of this readout is the $k$-deck of a word: the vector recording how often each word of length $k$ occurs as a subsequence. Two stored words are distinguishable from their readouts exactly when their $k$-decks differ, so the number $D_{q,k}(n)$ of distinct $k$-decks of words of length $n$ over an alphabet of size $q$ measures what a length-$k$ readout retains. We analyse the degrees of freedom remaining in a $k$-deck once all shorter decks are fixed. Within each class of words having prescribed letter multiplicities, the length-$k$ entries are confined to an affine subspace whose dimension is exactly the number of Lyndon words with the same multiplicities, which we give in closed form as a Möbius sum. Writing $L_q(j)$ for the number of Lyndon words of length $j$ over an alphabet of size $q$, we deduce the improved upper bound \[ D_{q,k}(n)=O\!\left(n^{E_q(k)}\right),\qquad E_q(k)=\sum_{j=1}^{k}j\,L_q(j)-1 . \] In the case of a binary alphabet this bound satisfies $D_{2,k}(n)=O\!\left(n^{4\cdot 2^{k-1}}\right)$. We then prove matching lower bounds in the first two nontrivial cases: $D_{q,2}(n)=\Theta\!\left(n^{q^2-1}\right)$ for every alphabet size $q$, and $D_{2,3}(n)=\Theta(n^{9})$ for the binary alphabet. The latter confirms, for $q=2$ and $k=3$, our conjecture that the upper bound has the correct degree for every $q$ and $k$.

We study the number of prototypes needed to represent Boolean functions by nearest-neighbour classification. There are two distinct settings: the prototypes may be arbitrary points of Euclidean space, or they may themselves be required to lie in the Boolean cube. For unrestricted prototypes, we strengthen a known lower bound for almost all Boolean functions. The bound applies simultaneously to nearest-neighbour voting rules with any number of voting neighbours, and substantially narrows the gap with the known general upper bound. We obtain a VC-dimension bound for classes with a bounded number of prototypes, and show that it is sharp in order in dimensions at least four. We then study Boolean prototypes, beginning with symmetric threshold functions. A connection with covering designs expresses the minimum number of prototypes at every threshold level exactly in terms of a covering number, and leads to further exact results for related monotone functions, including disjunctive extensions and a characterisation of when a representation with a single negative prototype is possible. For a uniformly random Boolean function, the Boolean nearest-neighbour complexity, as a proportion of the cube, is asymptotically close either to one half or to one, with explicit limiting probabilities. In particular, almost every Boolean function requires at least approximately half as many prototypes as there are points in the cube, and one half is the largest proportion for which such a lower bound holds. Finally, we consider arbitrary symmetric Boolean functions. Their Boolean nearest-neighbour complexity is closely approximated by a weighted vertex-cover problem on paths. As a consequence, a uniformly random symmetric function typically requires prototypes amounting to $11/20$ of the cube. This is much larger than the upper bounds known when the prototypes are allowed to lie anywhere in Euclidean space.

The notion of extremal subsets was introduced by Assaf, Dranowski, and González as a natural generalization of Demazure crystals. In this paper, we show that the crystal basis of a finite-dimensional irreducible module over a quantum group of finite type has a unique maximal proper extremal subset, whose complement is the set of elements that generate the whole crystal basis as an extremal subset. We conjecture that this complement consists of all nonzero elements obtained by successively applying the lowering Kashiwara operators to a distinguished element, which is constructed from the lowest element of the crystal basis. We prove this conjecture in type A.

A dominating set $D$ of a graph $G$ is a fair dominating set if every two vertices outside $D$ have the same number of neighbors in $D$, and the fair domination number $\mathrm{fd}(G)$ is the minimum cardinality of such a set. Caro, Hansberg and Henning, who introduced this parameter, proved that $\mathrm{fd}(G)<17n/19$ for every maximal outerplanar graph $G$ of order $n\geq3$, and asked whether this bound is asymptotically best possible. We show that it is not the case by proving $\mathrm{fd}(G)\leq(7n-3)/8<7n/8$ for every maximal outerplanar graph $G$ of order $n\geq3$, and we exhibit an infinite family of maximal outerplanar graphs with $\mathrm{fd}(G)/n\rightarrow7/9$, so that the best asymptotic constant lies between $7/9$ and $7/8$.

Let \(G=C(d_1,\ldots,d_n)=F_1\BoxProd\cdots\BoxProd F_n\) be a bipartite generalized hypercube with \(n\geq2\), all \(d_i\) even, and \(N=|V(G)|\geq8\), where \(F_i=K_2\) when \(d_i=2\), and \(F_i=C_{d_i}\) when \(d_i\geq4\). We prove the following exact strengthening of two-disjoint-cycle-cover vertex bipancyclicity. For every ordered pair of independent edges \(e,f\in E(G)\) and every even integer \(4\leq\ell\leq N-4\), the vertex set can be partitioned into two cycles \(J_1,J_2\) of lengths \(\ell\) and \(N-\ell\), respectively, with \(e\in E(J_1)\) and \(f\in E(J_2)\), if and only if \(G\) is not isomorphic to any \(K_2\BoxProd C_{2p}\) with \(p\geq3\). The graph \(C(2,2)\cong C_4\) is treated separately: it has no 2-DCC. Consequences that retain prescribed-edge information include ordinary edge bipancyclicity and the even \(k\)-ary \(n\)-cube specialization.

For $r \ge 0$, let $\mathcal P_r=\{\binom{r}{0},\binom{r}{1},\ldots,\binom{r}{\lfloor r/2\rfloor}\}$ be the set of distinct entries in row $r$ of Pascal's triangle. We study the least possible least common multiple of $n$ entries chosen from one row, with the row itself also free: $a(n)=\min_{r\ge0,\ S\subseteq\mathcal P_r,\ |S|=n}\operatorname{lcm}(S)$. Whereas the least common multiple of an entire row is given by a classical identity of Farhi, allowing both the row and the selected coefficients to vary creates a different optimization problem. We first recast the fixed-row problem exactly as a weighted prime-power exclusion problem, which explains how omitting a few coefficients can remove expensive prime-power contributions. Our main asymptotic result is $\log a(n)=2n+O\!\left(n\exp\!\left(-c(\log n)^{3/5}(\log\log n)^{-1/5}\right)\right)$ for some absolute $c>0$, so $a(n)^{1/n}\to e^2$. A two-band refinement further shows that every optimal row satisfies $r_n=2n+O\!\left(n\exp\!\left(-c(\log n)^{3/5}(\log\log n)^{-1/5}\right)\right)$, and that the minimum prefix defect of an optimal support is $o(n)$.

We prove the uniqueness conjecture of Fomin and Neville for their long mutation cycles in all ranks $n\ge4$. More generally, we show that every nucleus-extension quiver has a unique simple cycle in its labeled mutation graph. These quivers allow an acyclic part of arbitrary size outside the nucleus. Our proof uses three mutation statuses determined by a closed nucleus and the full subquivers obtained by deleting one part of its nucleus decomposition at a time from the ambient quiver. We analyze transitions between these statuses to prove that every mutation leaving the prescribed cycle is performed at an exit.

Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. For every $r\ge1$, we prove that a product-one-free sequence of length $2r+11$ over $E_2\times C_3^r$ cannot contain exactly $2r-1$ central terms. Thus the critical layer with twelve noncentral terms is excluded. The proof combines a relative moment criterion for abelian normal subgroups with a finite theorem in a symplectic four-space over $\mathbb F_3$. Under explicit subspace occupancy bounds, the family of balanced triangles that can actually be completed to a nonfull zero-sum block admits edge weights summing to one on every triangle. A dual cycle argument reduces this assertion to a potential condition on branching edges. All remaining configurations contain one of eleven minimal frames; two separately implemented exhaustive checks verify all their admissible extensions. The unrestricted check has 9544046 leaves. Complete source code and execution records are supplied. The exact value of $\mathsf{d}(E_2\times C_3^r)$ for arbitrary $r$ is not determined by this result.

An acyclic dicolouring of an oriented graph is a vertex partition in which every colour class and every bipartite subdigraph induced by two classes is acyclic. We first prove the Gallai–Roy-type bound $\vec{\chi}_a(D)\leq L(D)$, where $L(D)$ is the maximum order of a directed path. Let $r=\log_2(8/7)$. Bang-Jensen, Picasarri-Arrieta, and Yeo previously constructed tournaments of order $n$ whose acyclic dichromatic number is at least $n-(\frac8r)\log_2 n-\log_2\log_2 n$. We improve the leading logarithmic coefficient by a factor of two: for the uniform random tournament $\mathcal{T}_n$, asymptotically almost surely, $\vec{\chi}_a(\mathcal{T}_n)\geq n-\frac4r\log_2 n+\frac2r\log_2\log_2 n-O(1)$. Finally, if $m(k)$ denotes the minimum order of an oriented graph with acyclic dichromatic number at least $k$, tournament completion shows that the same minimum is obtained over tournaments. We prove $m(3)=5$ and $m(4)=7$ and classify the tournament witnesses at these minimum orders: there is one at order five and two at order seven.

The Pfaffian number of a graph is the minimum number of Pfaffians needed to obtain its perfect-matching polynomial by linear combination. In 2009, Norine conjectured that every Pfaffian number is a power of four. Miranda and Lucchesi disproved this conjecture in 2011 by constructing a graph of Pfaffian number six, and conjectured instead that every nontrivial Pfaffian number is even. We disprove their conjecture by proving that the Pfaffian number of $K_{3,3}\sqcup K_{3,3}$ is 13. We also construct a connected cubic bipartite matching-covered graph with Pfaffian number 13.

This work disproves the 1-leaky positive semidefinite edge-deletion conjecture and replaces it with a sharp theorem. For every leak level $\ell$ and edge $e$, one has $|Z^+_{(\ell)}(G)-Z^+_{(\ell)}(G-e)|\le 2$. More generally, if two graphs differ only on edges with both endpoints in $S$, their parameters differ by at most $|S|$. An endpoint-sensitive refinement recovers an increase of at most one whenever some minimum set for $G-e$ contains an endpoint of $e$. Both signs are sharp for every positive leak level. Joining two copies of $K_{\ell+1}$ by a bridge gives $Z^+_{(\ell)}(G)=2\ell$ and $Z^+_{(\ell)}(G-e)=2\ell+2$. For every $\ell\ge 2$, a connected clique-leaf pair of order $2\ell+3$ gives the opposite difference. The remaining positive-difference one-leak case is attained by connected graphs on nine vertices with $Z^+_{(1)}(H)=4$ and $Z^+_{(1)}(G)=6$. Explicit forcing sequences, fort certificates, and an exact verifier check the finite extremal example and stress-test the general results.

We construct a family of finite local rings whose annihilating-ideal graphs are naturally described by orthogonality of subspaces of $\mathbb{F}_2^n$. For $n=4$ we determine the clique and chromatic numbers exactly and obtain \[ \omega(\AG(R_4))=5<6=\chi(\AG(R_4)). \] Thus $\AG(R_4)$ is not weakly perfect, and the Behboodi–Rakeei conjecture fails for non-reduced commutative rings.

Let $q$ be a prime power. We give a construction of Butson Hadamard matrices $BH((F_q\times F_q,+),lcm(6,d))$ for any divisor $d>1$ of $q-1$. Most character values of the group ring elements corresponding to these matrices involve products of two Gauss sums over $F_q$ (with non-quadratic Gauss sums occurring) and this property distinguishes our result from all previously known constructions of group invariant Butson Hadamard matrices. In fact, our $BH((F_q\times F_q,+),lcm(6,d))$ matrices arise as a special case of a more general construction based on what we call “Butson-Jacobsthal functions” that resembles the construction of Hadamard matrices from Jacobsthal matrices.

For a finite abelian group $G$, let $\mu_2(G)$ be the minimum number of positive eigenvalues of an entrywise positive real symmetric group-circulant matrix whose entrywise square is positive definite. We prove that $\mu_2(G)\le5$ precisely when $|G|\le15$, and that $\mu_2(G)=6$ precisely when $|G|=16$ or $G$ is cyclic of order $18$, $20$ or $22$. For the noncyclic groups of order at most fifteen, the minimum is three for $C_2^2$ and five otherwise. If the positive Fourier support of such a matrix is $\{0,\pm r_1,\ldots,\pm r_d\}$, where $d\ge1$ and all $2d+1$ displayed elements are distinct, then there are integers $\alpha_1,\ldots,\alpha_d$, not all zero, such that $\sum_i\alpha_i r_i=0$ and $\sum_i|\alpha_i|\le\max\{5,d+2\}$. The bound five is sharp for $d=2$ and $d=3$.

The cube polynomial $C_G(x)$ generates the number of $k$-cubes on a graph $G$. As a subclass of partial cubes, the tope graphs of lopsided sets (LOPs) generalize daisy cubes and median graphs. In this paper, we prove that every tope graph of a LOP shares its cube polynomial with some daisy cube, thereby answering affirmatively a problem posed earlier by the authors. Furthermore, we present explicit expressions for the cube polynomials of tope graphs of LOPs: $C_G(x)=f_\mathcal{K}(x+1)$, where $f_\mathcal{K}(x)$ is the $f$-polynomial of the cubical complex $\mathcal{K}$ of $G$. Finally, we show that a class $\mathcal{G}$ of partial cubes is the class of tope graphs of LOPs if and only if the following equivalent conditions hold: (a) $\mathcal{G}$ is the maximal pc-minor-closed class such that every graph in $\mathcal{G}$ has the same cube polynomial as some daisy cube; (b) for every $G\in\mathcal{G}$, each antipodal subgraph of $G$ has the same cube polynomial as some daisy cube.

We introduce a combinatorial definition of chromatic symmetric functions for annular webs. We prove their symmetry by constructing a web analogue of the Shareshian–Wachs involution and show that they coincide with the symmetric functions associated to annular webs via Turaev's isomorphism. We then derive explicit formulas for their hook Schur coefficients. We also introduce web LLT functions, whose hook Schur coefficients admit positive Laurent-polynomial formulas. These formulas yield a combinatorial expression for the coefficients of the HOMFLY–PT polynomial of an annular web.

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, \Delta(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $\Delta(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $\Delta(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $\Delta(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $\Delta(9)$ remains open.

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $\pi/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.