Papers

Let $G$ be a connected graph with minimum degree at least two, positive Lin-Lu-Yau Ricci curvature, and no $K_{2,t}$ minor, where $t\ge 3$ is an integer. We first establish tight upper bounds on the maximum degree of $G$ by showing that $\Delta(G)\le 2t+3$ for $t=4$, $\Delta(G)\le 2t+4$ for $t\in\{5,6\}$, and $\Delta(G)\le 2t+2$ for all $t\ge7$. We then prove that $G$ has at most $10$ vertices for $t=3$, and at most $(t+1)\left(\Delta(G)^2(\Delta(G)-1)/2+1\right)$ vertices for all $t\ge 4$.

The theory of finite translation planes is intimately tied to the theory of spreads of PG(3,q), where a spread is a partition of PG(3,q) into $q^2+1$ pairwise disjoint lines. A traditional method of generating spreads begins with the regular spread and identifies sets of lines therein that can be replaced with other lines to create a new spread, such as reguli, Bruen chains, and nests. This work provides computational algorithms to enumerate all possible replaceable sets consisting of reguli, chains and nests in the regular spreads of PG(3,q) for $q \le 11$. Additional results include a refinement to Baker and Ebert's (q-1)-nests which expands that infinite family, and counterexamples to the long-standing conjecture that a t-nest of reguli in PG(3,q) must be replaceable if $t \le q$.

The Firefighter Problem models a spreading process (originally a fire, alternatively an infection or rumour, for example) on a graph. A defender saves a single vertex per turn; after each defence, the fire spreads to the unburned and undefended neighbours of all burning vertices. Deciding whether a strategy exists for the defender to protect some targeted number of vertices is computationally hard in graphs in general, but tractable in some restricted cases. Inspired by research into spreadable rabies vaccines for bats, we study a variant of the Firefighter problem in which defence also spreads. Some approximation results are already known for this problem; we provide algorithmic and hardness results, as well as containment results for the infinite $n$-dimensional Cartesian and strong grid graphs.

The recursive-line and signed Zarankiewicz numbers maximize the number of squares in augmentations of a maximum $C_4$-free base, subject to two sufficient irreducibility criteria. The count includes one square per base cell and one per selected pair of unused cells. We compare these parameters with the second-order number, which uses irreducibility itself. Every maximum $m\times n$ base admits a recursive-line augmentation with at least $mn/2-C\max(m,n)$ squares, for an absolute constant $C$. Combining this bound with a two-column extension of known fixed-width families, we show that all three parameters are asymptotically equivalent, uniformly as the larger dimension tends to infinity. For individual displays, a transfer graph shows that once the signed closure identifies every selected pair, the signed criterion is equivalent to irreducibility. We determine the second-order number for every six-column rectangle and give eventual exact formulas for all three numbers at widths seven, nine and eleven. We also prove signed and recursive-line equality at $8\times7$ and, together with earlier values, whenever the shorter side is at most six, except possibly at $14\times4$. The exact-value results are computer-assisted, using exhaustive enumeration, checked propositional refutations and symbolic certificates with a proved lifting argument for arbitrary lengths.

For each fixed $n$, we prove a positive lower bound for the diameter of a compact simple geodesic $n$-gonal monotile of the hyperbolic plane. We also prove a positive lower bound for the area of such a tile in a finite monohedral tiling of a closed hyperbolic surface, independent of the topology and metric. Both bounds become independent of $n$ for vertex-proper tilings, in which every genuine tile vertex belongs to at least three distinct tiles. This includes tilings by convex polygons. Reflex angles and non-edge-to-edge incidences are allowed. After qualitative proofs, we obtain explicit constants from a sharp gap estimate for packing polytopes with arbitrarily coupled nonnegative integer constraints. Exact corner balance applies on closed surfaces; in the plane, covering duality and ball counts give a boundary factor depending on the tile diameter. Following Zare, we give, for each integer $q\ge2$, a geodesic construction with $2q+3$ sides and diameter less than $3/q$, showing that side counts cannot be unrestricted without an additional condition. The closed-surface results extend to regular curved sides. The geometric constants are effective but are not claimed to be optimal.

In this paper we give necessary conditions for 4-class association schemes that are generated by double covers of strongly regular graphs. These conditions are applied to open cases for diameter 4 antipodal distance-regular graphs. Using these conditions we are able to show the nonexistence of four cases in the table of Brouwer, Cohen and Neumeier: $\{20,18,3,1;1,3,18,20\}$, $\{22,21,3,1;1,3,21,22\}$, $\{ 54,50,5,1;,1,5,50,54 \}$, $\{ 170,162,9,1;,1,9,162,170 \}$. More generally, we show there is no distance-regular graph with intersection array $\{k,b_1,b_2,1;1,b_2,b_1,k \}$ where $b_2 \neq 1$ and $\frac{ k b_1 }{4}(1+k+\frac{ k b_1 }{2b_2})$ is odd. Tables are also given for more general 4-class association schemes generated by double covers of strongly regular graphs.

In this paper, we compute the orders of upper triangular symplectic Renner monoids using admissible sets. We obtain both recursive and closed-form formulas for these orders. We then derive their exponential generating functions and establish connections between these functions and the Laguerre polynomials.

We study the realization problem for point schemes of skew polynomial algebras. We approach this problem combinatorially by translating it into the study of certain $3$-uniform hypergraphs, which we call point hypergraphs. We first give a homological criterion characterizing point hypergraphs. Using this criterion, we study hypergraphs obtained by deleting triangles from triangulations of connected orientable closed surfaces. Deleting exactly one triangle always gives a non-point hypergraph. When the triangulation has no separating nonfacial $3$-cycles, deleting either no triangles or at least two triangles gives a point hypergraph, and the one-triangle deletions are minimal non-point hypergraphs with respect to taking induced sub-hypergraphs. It follows that point hypergraphs cannot be characterized by finitely many forbidden induced sub-hypergraphs. Next, for each point hypergraph, we construct an affine moduli variety of skew polynomial algebras realizing it, and determine its dimension. For point hypergraphs arising from the above surface construction, we obtain an explicit dimension formula in terms of the number of vertices, the Euler characteristic, and the number of deleted triangles. Finally, we characterize point hypergraphs on six vertices in terms of a four-vertex local condition together with a single exceptional obstruction.

We show that there exists $C\in\mathbb{N}$ such that $K_t$-minor free graphs are $Ct$-colorable. The proof was found by GPT-6 Astra, following the directions by the authors.

The quantum $n$-coloring problem, given graph $G$, asks whether there exists a perfect quantum strategy for the $n$-coloring game of $G$. Previously it was known that quantum $n$-coloring is undecidable only for $n=3$. We extend this to all $n\geq 3$ with one elementary reduction. Letting $G'=(G\square K_3)\lor K_{n-3}$, we show that $G$ is quantum $3$-colorable if and only if $G'$ is quantum $n$-colorable.

For K-unstable toric varieties, Székelyhidi's optimal test-function $\Theta_{P}$ is a mysterious concave function over the moment polytope $P$, which gives the maximal destabilizer for K-stability, and encodes the limiting behavior of the divergent Calabi flow. As balanced norms quantize cscK metrics, we show that $\Theta_{P}$ can be quantized by the maximal destabilizers for Chow-stability, which are given by the shortest GKZ vectors, i.e., the least norm point on the secondary polytope for the set $P\cap k^{-1}\mathbb{Z}^{n}$. Properties and algorithm for general sGKZ vectors are given. Our result may provide a new method to detect the K-unstability of toric varieties.

Odd girth and the sum of the largest and smallest adjacency eigenvalues are two common indicators of how close a graph is to being bipartite. This paper investigates the asymptotic behavior of this eigenvalue sum under odd-girth constraints. We study the associated abstract moment problem over finite real lists satisfying the vanishing odd-moment identities imposed by odd girth at least $g$ and the standard second-moment bound for graph spectra. We prove that the supremum of $(\lambda_1+\lambda_n)/n$ over these lists lies between $(1-o(1))(\log g)^2/g^3$ and $(10/7+o(1))(\log g)^2/g^3$ as $g\to\infty$ through the odd integers. The upper bound applies in particular to graphs of odd girth at least $g$. For regular graphs of odd girth $g\geq5$, we obtain a stronger bound $(\lambda_1+\lambda_n)/n \leq 12/((g-1)(g-2)(g-3))$, which is optimal up to a constant factor.

A backbite on a Hamiltonian path of a graph $G$ adds an edge of $G$ from an endpoint $u$ of the path to a vertex $v$ and deletes an edge of the path at $v$ so that the result is again a Hamiltonian path. We study the connectivity of the reconfiguration graph $R(G)$, whose vertices are the Hamiltonian paths of $G$, with two adjacent when a single backbite carries one to the other. We prove that $R(G)$ is connected whenever $G$ contains the square of a Hamiltonian path as a spanning subgraph, and that any two Hamiltonian paths can be reconfigured using $O(n^2)$ backbites. Combining the theorem with known results on spanning squared paths gives connectivity for every graph with $\delta(G)\ge\lceil\frac{2n-1}3\rceil$, for the circulants $\mathrm{Circ}(n;S)$ with $a,2a\in S$ for some $a$ coprime to $n$, and, with high probability, for the random graph $G(n,p)$ with $p\ge c/\sqrt n$, for a sufficiently large constant $c$. In the other direction, graphs with arbitrarily high connectivity may have disconnected reconfiguration graphs. For every $n\ge8$ and every $1\le m\le\lfloor n/2\rfloor-3$ we construct a graph on $n$ vertices with $\kappa=\lambda=\delta=m$ whose reconfiguration graph is disconnected, with at least $(n-2m-4)!$ components. We conjecture that $\delta(G)\ge n/2$ suffices for connectivity.

In 1949, De Bruijn and Erdős considered infinite stick-breaking sequences. In this work, we consider finite sequences and prove that an infinite sequence that is due to De Bruijn, Erdős, and Toulmin is in some sense unique.

Let $f(d)$ be the smallest integer such that every connected bridgeless graph of diameter $d$ admits an orientation with directed diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\left\lceil{d^2}/{2}\right\rceil+d\le f(d) \le 2d^2+2d$. In this paper, we prove that $f(d)\le \left\lceil d^2/2\right\rceil+6d$, which shows $f(d)= d^2/2+\Theta(d)$ and determine the leading quadratic coefficient of $f(d)$. The key method of our proof is to construct a central subgraph $H$ such that it admits a strong orientation of diameter $O(d)$ and the distance between $H$ and every vertex outside $H$ is at most $ d/2 $.

The two-block odd partition function \(a(n)\) is the signed enumeration of partitions into exactly two distinct part sizes, each occurring an odd number of times. By separating the underlying representations according to the \(2\)-adic valuations of the two part sizes, we obtain a signed decomposition, completed by two classical theta-function evaluations, which explains the structural relation \(a(2^km)=a(m)+(2^{k-1}-1)\sigma(m)\), where \(m\) is odd, \(k\ge1\), and \(\sigma(m)\) is the sum-of-divisors function. Combined with a previously established arithmetic formula for \(a(n)\), this relation yields congruences along the dyadic progressions \(2^k(4n+3)\), as well as the families \(a\!\left(2^\alpha3^\beta(12n+11)\right)\equiv0\pmod3\) and \(a\!\left(2^\alpha(18n+15)\right)\equiv0\pmod3\). We also use a factorization of the generating function of \(a(n)\) involving \(\operatorname{pod}(n)\), the number of partitions in which odd parts are distinct and even parts are unrestricted, to establish a uniform reduction modulo every odd prime of the associated auxiliary series. The resulting theta-weighted sums involving \(\operatorname{pod}(n)\) are congruent to \(a(n)\) and therefore admit explicit evaluations involving the sum-of-divisors function and the character divisor sum associated with the nonprincipal Dirichlet character modulo \(4\). As special cases, we obtain infinite families of weighted congruences modulo \(3\), \(5\), and \(13\).

We prove the Baker–Rumely conjecture that the tau invariant of a metrized graph admits a positive lower bound proportional to its total length, with an absolute constant. We also construct simple cubic metrized graphs whose normalized tau invariants tend to $59/7260<1/108$, disproving the proposed universal constant $1/108$. The lower bound is independent of the genus, the number of edges, and the distribution of edge lengths. Its proof combines a second-moment inequality for Euclidean lattices with a partition of the edge coordinates of a cycle lattice into three independent sets. The counterexamples have only two edge lengths and admit an elementary resistance calculation. Through the tropical moment identity, the lower bound also gives a uniform estimate for the non-archimedean terms in height formulas for Jacobians.

The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre–Pólya class. Brändén [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre–Pólya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\lambda_n)_{n\ge0}$. In this paper, we prove that the sequence $(\lambda_n)_{n\ge0}$ is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.

Let $\lambda_1(n,\Delta)$ be the maximum adjacency spectral radius among connected nonregular simple graphs of order $n$ and maximum degree $\Delta$. Using effective resistance bounds, explicit comparison graphs and a one-dimensional Wirtinger inequality, we prove that, for every fixed integer $\Delta\ge3$, $\lambda_1(n,\Delta)=\Delta-c_\Delta\pi^2/(4n^2)+O_\Delta(n^{-5/2})$ as $n\to\infty$, where $c_\Delta=\Delta-1$ for odd $\Delta$ and $c_\Delta=2(\Delta-2)$ for even $\Delta$. This proves the asymptotic conjecture posed by Liu [J. Combin. Theory Ser. B 169 (2024), Conjecture 7.1].

Let $G$ be a graph. Under an injective edge coloring of $G$, any two edges at distance $2$ or belonging to a common triangle receive distinct colors. The injective chromatic index of $G$, denoted by $\chi'_{inj}(G)$, is the smallest integer $k$ such that $G$ admits an injective edge coloring with $k$ colors. Let $\Delta$ be the maximum degree of $G$. Ferdjallah et al.~proved that $\chi'_{inj}(G) \le 2(\Delta-1)^2$. In this paper, we improve this bound by proving that $\chi'_{inj}(G) \le 2(\Delta - 1)^2 -\Delta + 3$ if $\Delta \ge 4$.

Let $r\geq 2$ be an integer. An $r$-partite graph $\Gamma$ with vertex partition $V_1,\ldots,V_r$ is $2$-balanced if there exists a positive integer $n$ such that $n\leq |V_i|\leq 2n$ for every $i\in[r]$. For an integer $\ell\geq3$, let $C_\ell$ denote the cycle of length $\ell$, and define $\hat{\delta}(\Gamma)=\min\{d_\Gamma(v,V_i)/|V_i|:i\in[r],\ v\in V(\Gamma)\setminus V_i\}$. Let $\hat\delta^r_{C_\ell}$ denote the $C_\ell$-decomposition threshold for $2$-balanced $r$-partite graphs, that is, the least nonnegative real number $\delta$ such that, for every $\varepsilon>0$, there exists $n_0$ such that every $C_\ell$-divisible $2$-balanced $r$-partite graph $G$ with $\min_i|V_i|>n_0$ and $\hat\delta(G)\geq\delta+\varepsilon$ admits a $C_\ell$-decomposition. We prove that $\hat\delta^{r}_{C_4}=\frac{2}{3}$ and $\hat\delta^{r}_{C_{2k}}=\frac{1}{2}$ for every $r\geq2$ and every $k\geq3$.

For an odd prime power $q$, a BLT-set $\mathcal B$ is a set of $(q+1)$ points of $\mathcal Q(4,q)$ such that every point of $\mathcal Q(4,q)$ is collinear with at most two points of $\mathcal B$. We complete the classification of BLT-sets of $\mathcal Q(4,q)$ admitting a non-solvable transitive automorphism group. Nelson and Penttila completed the classification of BLT-sets $\mathcal B$ with a non-solvable transitive automorphism group acting irreducibly on the underlying $5$-dimensional vector space. In this paper, we prove that a BLT-set with a non-solvable transitive full automorphism group acting reducibly on the underlying $5$-dimensional vector space is either a classical BLT-set or a Kantor semifield BLT-set.

Motivated by the EPPA problem for finite tournaments,we consider various extension properties for partial permutations. We show that for any set $\Pi$ of prime numbers, the $\Pi$-extension property is equivalent to the $\Pi$-LERF. As a consequence of known results, the odd-extension property is then equivalent to the EPPA for finite tournaments. To study the $\Pi$-extension property, we reformate the property using the concepts of EP problems and $\Pi$-solutions. We then show that the existence of a $\Pi$-solution for an EP problem depends entirely on its fundamental group. In particular, when the fundamental group is trivial or cyclic, the EP problem has a $\Pi$-solution for any $\Pi$. These extend some known results of Huang, Pawliuk, Sabok and Wise [HPSW19]. We give examples of EP problems without nilpotent-solutions; they witness that the $p$-extension property and the $p$-LERF fail for any prime $p$. Then we consider a special kind of EP problems whose fundamental groups have two generators. We show that any 1-weakly wandering problem has a $\Pi$-solution for any $\Pi$. For 2-weakly wandering problems, we show that they all have odd-solutions and we completely characterize those without nilpotent-solutions.

We prove several conjectures from the On-Line Encyclopedia of Integer Sequences about the lower and upper Wythoff sequences and the Fibonacci word. Among them are two of Kimberling's three conjectures on the number of ways to write $n=\lfloor h\varphi\rfloor+\lfloor k\varphi^2\rfloor$ with $h,k\ge1$ (A259598): exactly one way if and only if $n+1=2F$ for a Fibonacci number $F\ge2$, and exactly two ways if and only if $n+1\ge7$ is a Lucas number. The third conjecture, that no way exists if and only if $n+1$ is a Fibonacci number, was proved earlier by Kawsumarng et al. We also observe that Kimberling's conjecture on the gaps of the sums of two distinct terms of A003622 and of their complement (A333308, A333309) follows, after a shift by $2$, from earlier Walnut results of Shallit and of Bosma et al. on A260317, and we re-verify it. Next, we prove Kimberling's five 2025 conjectures on the gaps between positions where the Fibonacci word and the "Lucas word" take prescribed values (A383423-A383427). Finally, we prove a conjecture of Mathar on A285383, and we point out that a conjecture of Schmidt on A003250 follows from theorems of Carlitz, Scoville and Vaughan (1973); we also confirm it with Walnut. Most proofs are decision procedures run in the free prover Walnut, and we supply the complete command file.

We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight $X$- and $Z$-type Pauli shifts, analogous to small phase-space displacements in continuous-variable systems. This structure is approximately respected by nuclear-spin noise and captured by the Lee metric, motivating a quantum extension of classical Lee-metric coding theory. We develop such a formalism for stabilizer codes over $\mathbb{Z}_q$ for arbitrary $q$, using joint and separate Lee metrics for the $X$- and $Z$-components of errors. For qubits, the joint metric counts $Y$ errors twice and can yield codes that detect and correct $X$ and $Z$ errors with fewer physical qubits than codes designed for the conventional Hamming metric. We discuss Lee-weight spreading under Clifford gates and qubitize codes over $\mathbb{Z}_4$ via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on $n$ qudits, we prove that the Lee distance cannot exceed $O(n)$, uniformly in $q$, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-$q$ qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical $q$-ary “helical repetition codes” have exponentially long memory times at fixed temperature for sufficiently large $q$ (that grows with system length), which can be understood as spontaneous symmetry breaking at finite temperature, even in local one-dimensional models. Hypergraph products of these helical repetition codes provide local two-dimensional quantum codes that inherit self-correction for $Z$ errors, but self-correction for $X$ errors remains an open question.

We modify a construction of Bradač \cite{Bradac26} using a vertex ordering to give, for every prime power $q$, an explicit $K_5$ free graph with more than $q^7$ vertices and independence number below $600q^4$. This shows $$r(5,t)=\Omega(t^{7/4}).$$ Additionally, pruning a graph defined by the Coulter and Matthews polynomial $X^{14}$ \cite{CM}, we give a deterministic polynomial time construction proving $$r(5,t)=\Omega\bigl(t^{20/11-\varepsilon}\bigr)$$ for every fixed $\varepsilon>0$. These results improve the constructive lower bound $\Omega(t^{5/3})$ of Kostochka, Pudlák and Rödl \cite{KPR}.

Benjamini asked whether every configuration of points on the real line that is in equilibrium under the inverse-square repulsive force must be an arithmetic progression. Georgakopoulos and Kolountzakis proved this when some gap between consecutive points has maximal or minimal length, and described the general (aperiodic) case as open. We show that the answer is yes. More generally, let $1<s\le2$, and let $X\subset\mathbb{R}$ be a locally finite set with at least two points such that, for every $x\in X$, the total force $\sum_{y\in X\setminus\{x\}}|y-x|^{-s}$ is finite and the net force $\sum_{y\in X\setminus\{x\}}\operatorname{sgn}(y-x)\,|y-x|^{-s}$ is zero. Then $X$ is an arithmetic progression. No a priori assumption on the gaps is needed. Subtracting the equilibrium equations of two consecutive points shows that the gaps $g_n$ form a positive harmonic function for an explicit reversible random walk on $\mathbb{Z}$ with long-range jumps. Equilibrium also bounds the ratio of consecutive gaps, by $1.5386\ldots$ when $s=2$. With this bound, an energy estimate shows that the Doob transform of the walk by $g$ is recurrent. Since $1/g$ is a positive harmonic function of the transformed walk, it is constant.

Braces play a fundamental role in matching theory, as they, together with bricks, constitute the basic building blocks of matching covered graphs in the tight cut decomposition. A brace is minimal if deleting any edge from it results in a graph that is not a brace. We prove that every perfect matching of a minimal brace of order at least six contains an edge whose ends both have degree three. Consequently, every such brace has at least three edges with this property. Combining this result with the forest structure induced by the noncubic vertices, we show that every minimal brace $G$ of order $n\ge6$ and size $m$, other than $K_{3,3}$, satisfies $n_3(G)\ge\max\left\{8,\left\lceil\frac{2n+8}{5}\right\rceil, \left\lceil\frac{m-n+4}{2}\right\rceil\right\}$, where $n_3(G)$ is the number of cubic vertices of $G$. Finally, we characterize the graphs attaining the constant lower bound: a minimal brace has exactly eight cubic vertices if and only if it is isomorphic to $B_8$, $B_{10}$, $Q_{10}^{+}$, or $Q_{12}$.

In Divisor Nim, a player removes from one heap a positive number of stones that divides the size of every other heap. We prove that the Sprague–Grundy (SG) value satisfies $g(P)\le 2h_{\min}+\lfloor\log_2h_{\min}\rfloor$, where $h_{\min}$ is the smallest nonempty heap, independently of the number and sizes of the other heaps. The proof starts with odd heaps and classifies removals by their factors of $2$. The largest heap in a starting board then supplies an SG bound valid throughout every successor chain. When one heap varies, we sharpen Morisawa's eventual-period bound: the smallest fixed heap controls the SG value range, the number of consecutive values compared, and an increase in the varying heap that repeats legal moves. The remaining dependence on the fixed board lies in the SG periods of direct successors. Using the largest fixed heap, we derive an explicit common period for all these recursive layers. Its possible prime divisors belong to a fixed finite set, independent of heap count; their exponents remain controlled by the total fixed stones. Finally, joining boards under one shared divisor rule preserves and can tighten the SG ceiling. Minimum 2-adic depth and its count parity determine every joined outcome, and a join with at least two odd heaps has exact SG value $0$ or $1$. We close with two conjectures: $g(P)\le2h_{\min}$, and that the least eventual SG period, when one heap varies, divides $2\operatorname{lcm}(1,\ldots,h_{\max})$. Here $h_{\max}$ is the largest fixed heap, excluding the varying heap. The second claim would remove heap-count dependence from the period bound as well.

A Moore graph simultaneously asks for two things that normally pull in opposite directions: every vertex should have many neighbors, while every pair of vertices should remain close. For diameter two and degree $k$, the Moore bound is $k^2+1$, and equality forces a remarkably rigid graph: it is strongly regular with parameters $(k^2+1,k,0,1)$. Hoffman and Singleton showed that such a graph can exist only for $k=2,3,7$, and possibly 57. The degree-57 case on 3,250 vertices remains open. This paper is not an attempt to settle that open problem. Instead, we ask a more algorithmic question: How efficiently can the defining Moore structure itself be searched exactly? We begin from a human-developed exhaustive search based on the radius-two Moore tree. Leaves fall into $k$ groups of size $k-1$, and the edges between every pair of such groups form a perfect matching - a permutation. For efficiency, a partial construction is abandoned immediately when it creates a triangle or quadrangle. We then describe a sequence of exact algorithm-engineering improvements developed through a human-guided AI process and accepted only after mathematical checking and controlled regression tests. The main steps are reversible short-cycle forbid counters, bit-parallel domains, minimum-remaining-value branching, singleton propagation, matching-based AllDifferent filtering, balanced replayable parallel frontiers, residual-symmetry orbit pruning, and canonical deduplication of frontier states. On the principal $k=9$ benchmark, stronger symmetry cuts wall time from 47.6 to 17.9 seconds on a six-core laptop setup. Further parallelization speedup is reported separately; on a 16-core/32-thread machine, median wall time is reduced to 5.28 seconds. Degrees 10 and 57 are used as stress tests, and a five-minute degree-57 run uses only 161 MiB peak resident memory, showing that state size is modest.