Papers

We address Questions~8.1 and~8.4 of Coron–Ferroni–Li \cite{coron2025chow}. For the first part of Question~8.1, which asks whether every rank-uniform EL-shellable poset admits a rank-uniform EL-labeling, we construct a counterexample of rank $4$ and prove that this rank is minimal. For the second part of Question~8.1, concerning whether every rank-uniform EL-shellable poset is UMEL-shellable, we construct a counterexample of rank $3$. We also develop a general criterion for constructing UMEL-shellable posets of rank $3$. As an application, we give an affirmative answer to Question~8.4 of Coron–Ferroni–Li.

A polyhex is an edge-connected set of n cells of the regular hexagonal tiling, where each cell has area one. We prove that the convex hull of a polyhex has area at most (1/6)*ceiling(n^2 + 14n/3), and we show that some polyhex reaches this bound for every n. This proves a conjecture of Kurz from 2008, which asked for the weaker bound (1/6)*floor(n^2 + 14n/3 + 1). The two bounds differ exactly when 3 divides n. We checked the upper bound in the Lean 4 proof assistant with the Mathlib library. We also report a computation over all polyhexes with at most 12 cells, which shows that for these sizes only one shape reaches the maximum, up to rotation and reflection.

Let $\lambda\succ\gamma\succ\mu$ be equal-degree nonnegative integer exponent vectors with at most $n$ parts, let $J_\eta^{(n)}$ be the labeled symmetric orbit sum associated with $\eta$, and set \begin{equation*} P_n=J_\lambda^{(n)}+J_\mu^{(n)}-2J_\gamma^{(n)}. \end{equation*} For every $n\ge4$, we prove that global nonnegativity of $P_n$ on the positive orthant is equivalent to nonnegativity on the fixed two-variable section \begin{equation*} (x,y,1,\ldots,1),\qquad x,y>0. \end{equation*} Thus the positivity of this family is determined by a fixed two-dimensional section, independently of both degree and ambient dimension. In the injective hierarchy $C_k$ introduced below, the hypothesis is $C_2\ge0$, global positivity is equivalent to $C_{n-1}\ge0$, and the main step is the stable order-raising implication \begin{equation*} C_k\ge0\Longrightarrow C_{k+1}\ge0, \qquad 2\le k\le n-2. \end{equation*} For $n\ge5$, this rank-two hypothesis is strictly weaker than positivity on the normalized full collision wall, which corresponds to $C_{n-2}\ge0$. The result complements the preceding three-variable theorem Collision Positivity for Three-Variable Symmetric Monomial Inequalities: A Complete Boundary Criterion, which proves the exceptional rank-one step $C_1\Rightarrow C_2$. The proof here is logically independent: the local three-label ingredients needed in higher rank are established directly. The main tools are a Green second-compound reserve, fixed-union exchange on adjacent subset ranks, a rank-three zero/one/two-mark contraction, and terminal boundary recombination. The theorem extends to nonnegative rational exponents by clearing denominators.

We introduce the directed cyclic difference inventory $D_n(\mathbf{m})$ to study Latin Eulerian numbers restricted to row-reorderings of the cyclic Latin square, offering an orientation-sensitive refinement of the prescribed-edge-length Hamiltonian path problem. We establish exact enumeration formulas, symmetries, and realizability obstructions for this inventory. Furthermore, we reduce the cyclic total-ascent statistic directly to endpoint-refined Eulerian statistics via the identity $\Sigma(L_\pi) = n \operatorname{asc}(\pi) + \pi(1) - \pi(n)$, yielding a closed-form expression for the cyclic Latin-Eulerian polynomial.

The Meir–Moser rectangle-packing problem asks whether all rectangles with side lengths \(1/n\) and \(1/(n+1)\), for \(n\ge1\), can be packed into the unit square with pairwise disjoint interiors. We establish a tail version of this problem. Let \(R_n\) denote the rectangle with these side lengths. We prove that there exists an integer \(m_0\) such that, for every \(m\ge m_0\), the family \(\{R_n:n\ge m\}\) admits a packing, by translations and right-angle rotations, into a square of side length \(m^{-1/2}\), with pairwise disjoint interiors. The area of the square equals the sum of the areas of all the rectangles. The geometric construction recursively decomposes rectangular gaps, while local randomized quotas and random permutations assign subsequent integer indices. We separately control the total area of waiting gaps and the assignment load at each index. The proof is organized in six steps: a finite-prefix reduction, geometric row decompositions, an area bootstrap, a sharp source-load estimate, control of the actual adaptive construction, and a compactness limit. For every finite time horizon, the probability of failure has a bound that is independent of the horizon and can be made arbitrarily small. The adaptive step uses a permanent load ledger, actual fresh height queries, and a one-sided comparison with a frozen source experiment. Compactness then yields an infinite packing. The final packing statements have been checked in Lean 4. A sufficient threshold is \(m_0=10^{1000}\). This result applies only to sufficiently late tails and does not resolve the original Meir–Moser rectangle-packing problem for the full sequence starting at \(n=1\), which remains open.

A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in $(1,2\cos(\pi/9)]$ and one in $[-2\cos(\pi/9),-1)$, and hence is not integral. This settles a conjecture of Braga, Del-Vecchio and Rodrigues (2021) on integral generalized sun graphs. The argument is matrix-theoretic: adjoining a terminal path on three new coordinates to an arbitrary real symmetric matrix produces the same spectral obstruction, and the positive interval above is optimal in this generality. We then disprove a second conjecture of the same authors, which asserts that the cycle of an integral generalized sun graph other than a cycle has length divisible by four. The graph obtained from a hexagon by attaching $6,6,12,6,6$ pendant vertices to five of its six vertices is integral and has $42$ vertices. We show that it is the smallest member of an infinite family governed by the Pell equation $x^{2}-2k^{2}=-7$, and we compute in closed form the characteristic polynomial of the analogous graphs on an arbitrary even cycle. Integrality within this family forces the cycle to be a square or a hexagon, and the square case yields a second infinite family governed by the Pell equation $k^{2}-2c^{2}=1$.

We study nonpreemptive scheduling on a single machine with release dates, due dates, positive job weights, and a common processing time. The objective is to minimize total weighted tardiness. Although closely related equal-processing-time problems admit polynomial-time algorithms, the complexity of this problem has remained open in the literature since 2010. We prove that its decision version is strongly NP-complete, even when every job can meet its due date if processed immediately upon release. The reduction is from unweighted MAX-CUT and uses a quadratic number of jobs with polynomially bounded numerical data. Its main ingredient is a constructive normalization theorem that converts every sufficiently inexpensive feasible schedule into a binary choice for each graph vertex; after normalization, total weighted tardiness equals a constant minus a scaled cut value. We also give a deterministic polynomial-time phase-grid assignment algorithm for the shifted objective $\Phi=F+p\sum_jw_j$, where $F$ is total weighted tardiness. The algorithm enumerates at most $N$ release-date residues modulo $p$, solves one minimum-cost assignment problem for each residue, and returns the best phase-grid schedule. It runs in $O(N^5)$ arithmetic operations and achieves the tight ratio $3/2-1/(2N)$ for this algorithm. Because the added term $p\sum_jw_j$ is independent of how the jobs are scheduled, the shifted and original objectives have exactly the same optimal schedules. However, the approximation guarantee applies to the shifted objective; for the original objective, the analysis provides an additive bound. Thus, the paper both resolves the long-standing complexity question and provides a complementary worst-case guarantee for the phase-grid assignment algorithm.

Let $\rho(n,d)$ denote the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $d$. A graph attaining this maximum is called an extremal graph. Liu [J. Combin. Theory Ser. B, 2024] determined the extremal graphs for $d=3,4$ and formulated two conjectures for general $d$. For each fixed odd integer $d\ge3$, the conjectures assert that: (1) $\displaystyle\lim_{n\to\infty}n^2\bigl(d-\rho(n,d)\bigr) =(d-1)\pi^2/4$. (2) For all sufficiently large $n$, the degree sequence of every extremal graph is $(d,\ldots,d,d-1)$ for odd $n$ and $(d,\ldots,d,1)$ for even $n$. We prove the first conjecture for every fixed odd $d\ge3$ and, more precisely, obtain the asymptotic expansion \[ \rho(n,d) =d-\frac{(d-1)\pi^2}{4n^2} +\frac{(d-1)^2\pi^2}{4n^3} +O_d(n^{-4}) \qquad(n\to\infty). \] We further prove the second conjecture for every fixed odd $d\ge3$.

We study square-complementary graphs $G$ (satisfying $G^2 \cong \overline{G}$). We show that in such graphs, no two vertices have comparable closed neighborhoods. This implies that nontrivial square-complementary graphs have no simplicial vertices and are not chordal, thus solving two open problems posed in Discrete Mathematics 327 (2014) 62-75. We also show that no nontrivial square-complementary graph is distance-hereditary.

Let $\pi=\pi_1\cdots \pi_n$ be a permutation. We say that $\pi$ is $run$-$sorted$ if $\pi_1=1$ and the entries immediately following the descent positions of $\pi$ form an increasing sequence. Let $\mathcal{R}_n$ denote the set of run-sorted permutations of length $n$, which has cardinality given by the Bell number $B_{n-1}$ for all $n \geq 1$. In this paper, we consider the joint distribution $A_n(q,u)$ on $\mathcal{R}_n$ for the parameters tracking the numbers of inversions and runs leading to a new polynomial generalization of the Bell numbers. Among our results, we find a general recurrence for $A_n(q,u)$, from which one may derive explicit formulas for the total numbers of inversions or runs in all the members of $\mathcal{R}_n$ as well as for the sign-balance on $\mathcal{R}_n$ of either parameter. A simple expression for the Eulerian generating function for $A_n(q,1)$ may be found upon making use of Gessel's $q$-exponential formula which can be extended to general $u$. Finally, a formula is found by a direct argument for the maximum number of inversions within a member of $\mathcal{R}_n$.

Pattern avoidance is a central topic in the study of permutations. Parking functions provide a natural setting in which to ask analogous questions. In this paper, we study parking functions whose parking permutations avoid the patterns $1234$ and $2134$, obtaining an analog of the Schröder numbers for parking functions. To enumerate these objects, we develop an enumeration scheme that runs in polynomial time and adapt the prefix-based framework introduced by Zeilberger in 1998, and later improved by Vatter, by instead using “suffixes.”

Rooted phylogenetic networks are widely used to represent the evolution of species that have undergone reticulate processes. However, these networks can be highly non-planar, making them more difficult to visualise and interpret than evolutionary trees. In this paper, we investigate planarity properties of galled networks, an important subclass of phylogenetic networks. We show that all planar galled networks are necessarily upward planar. Furthermore, by leveraging recent results on planar phylogenetic networks, we provide three characterisations for each of the outerplanar and terminal planar galled network classes in terms of forbidden vertex configurations, forbidden directed subgraphs, and forbidden structures in their associated underlying undirected graphs. These results contribute to a deeper understanding of the structural properties of galled networks and may inform future methods for their construction and visualisation.

This paper is devoted to studying permutation binomial $F_{r,a}(X)=X^r(X^{q-1}+a)\in\mathbb F_{q^e}[X]$ with $a\in\mathbb F_{q^e}^*$. We present a complete characterization for $F_{r,a}$ to be a permutation of $\mathbb{F}_{q^e}$. This yields a complete proof of the conjecture proposed by Masuda–Rubio–Santiago \cite{masuda2022permutation}. More precisely, we show that \(F_{r,a}\) permutes \(\mathbb F_{q^e}\) if and only if $(-a)^{(q^e-1)/(q-1)}\ne1, \gcd(r,q-1)=1$, and there exists \(1\le h<e\) with \(\gcd(h,e)=1\) such that $r(q^h-1)\equiv q-1\pmod{q^e-1}.$

The Erdős-Gyárfás conjecture posits that every graph with minimum degree at least three contains a cycle of length some power of two. We prove a few simple structural properties for any minimal counter-example to this conjecture. In particular, the fraction of its vertices of degree three must be greater than $2/3$, thus improving on the prior bound of $4/7$ (Carr, 2026). Furthermore, it is either biconnected or the $1$-clique-sum of two biconnected graphs. By exploiting some of these properties, we were able to verify the conjecture for every graph of order at most $40$, every bipartite graph of order at most $66$, and every cubic graph of order at most $48$.

For a fixed integer $k\ge 2$, let $W_{2k+1}=K_1\vee C_{2k}$ be an odd wheel graph. The fixed-size spectral extremal problem aims to determine \[ \operatorname{spex}(m,W_{2k+1}):=\max\{\rho(G): e(G)=m,\ G \text{ is } W_{2k+1}\text{-free}\}, \] where $\rho(G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [12] proposed the following conjecture: When $m-\binom{k}{2}$ is divisible by $k$ and $m$ is large, every $W_{2k+1}$-free graph of size $m$ satisfies \( \rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2} \) with equality precisely for $K_k\vee qK_1$. For nonzero residue class, Yu, Zhang, and Zhang [13] proposed the following conjecture: Let $r$ be a nonzero remainder when $m-\binom{k}{2}$ is divided by $k$ and $m$ is large. Then $S_{k,m}$ is the unique graph among $W_{2k+1}$-free graphs of size $m$ having maximum spectral radius, where $S_{k,m}$ is obtained from $K_k\vee qK_1$ by adding a vertex $z$ and joining it to exactly $r$ vertices of the $K_k$. Very recently, Fang, Zhai and Zhang [4] confirmed the Yu–Li–Peng conjecture for $k\ge 2$. Chen, Gao and Li [2] confirmed Yu-Zhang-Zhang conjecture for $k\ge 3$. When $k=2$, then $W_5=K_1\vee C_4$. For large odd $m$, determining $\operatorname{spex}(m,W_5)$ is still open. In this paper we address the odd-size problem. Our result disproved Yu-Zhang-Zhang conjecture for $k= 2$. In our proof, a universal defect bound shows that only $O(1)$ edges can lie outside the dense core. Perron localization then reduces this to at most one edge. A discrete defect inequality forces the complete bipartite crossing and quantizes the two matching deficiencies. Exact quotient-polynomial comparisons eliminate the remaining cross-edge and odd–odd candidates.

Lecture notes from a graduate course given by Michael Krivelevich at the School of Mathematical Sciences of Tel Aviv University in the spring semester of 2026. Topics covered include: phase transition and the giant component in $G(n,p)$; long paths and cycles in supercritical and sparse random graphs; thresholds for connectedness and perfect matching; general model of a random subgraph of a finite graph; phase transition and the giant component in the random hypercube; polynomial diameter of the giant component; perfect matchings in the random hypercube.

Graphical discreteness is not preserved under quasi-isometry in general. We prove that, for every finite thick generalized $m$-gon $\Gamma$ with $m\geq 3$, graphical discreteness is nevertheless constant on the class of finitely generated groups quasi-isometric to the right-angled Coxeter group $W_\Gamma$: such a group $\Lambda$ is graphically discrete if and only if $\Gamma$ is nonflexible, meaning that no nontrivial graph automorphism of $\Gamma$ fixes a closed star pointwise. The proof combines quasi-isometric rigidity of Fuchsian buildings with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. For finite-rank Coxeter systems, we also show that the automorphism group of the standard Cayley graph is compact-by-discrete if and only if it is discrete, equivalently, if and only if the defining diagram is nonflexible. We further obtain obstructions to graphical discreteness from one-vertex doublings, classify compact-by-discreteness for standard chamber graphs of graph products of finite groups, and derive applications to finite projective planes and to right-angled Coxeter groups with Menger curve boundary. Our methods do not settle whether nonflexible finite thick generalized polygons exist.

Let $p$ be a prime and $\mathbb{F}_q/\mathbb{F}_r$ a finite field extension with $q=p^m$ and $r=p^s$. For any $k\mid q-1$, we consider $r$-ary irreducible cyclic codes (ICC) of the form $C(k,q/r) = \{(Tr_{q/r}(\gamma \omega^{ik})_{i=0}^{n-1})\}_{\gamma \in \mathbb{F}_r}$, with $\omega$ a primitive element of $\mathbb{F}_q$ and $ n= \tfrac{q-1}{k}$, and generalized Paley (GP) graphs $\Gamma(k,q) = Cay(\mathbb{F}_q, \{ x^k : x \in \mathbb{F}_q^* \})$. We show that there is a simple closed formula relating the weight distribution of $C(k,q/r)$ with the spectrum of $\Gamma(k_r,q)$, where $k_r=\gcd(k, \frac{q-1}{r-1})$. Then, we give $Spec(\Gamma(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. Finally, we give the weight enumerators of irreducible cyclic codes associated with Hamming GP-graphs.

In this paper we determine the Ramsey number $R(3,3,k)$ up to a constant factor, showing that $$R(3,3,k) = \Theta\bigg( \frac{k^3}{(\log k)^2} \bigg).$$ The proof of the lower bound combines the Hefty-Horn-King-Pfender construction for $R(3,k)$ with the method of Alon and Rödl. Using the same proof, we also determine the $r$-colour Ramsey numbers $R_r(3,\ldots,3,k)$ up to a constant factor for every fixed $r \geqslant 3$.

We prove an averaging formula for the Ehrhart polynomial of a cyclic polytope whose vertices are given by integer parameters on the moment curve. More precisely, its Ehrhart polynomial is the average of the Ehrhart polynomials of an explicitly constructed family of lattice zonotopes. Since lattice zonotopes are magic positive, this formula implies magic positivity for these cyclic polytopes. Consequently, their $h^\ast$-polynomials are real-rooted, and their $h^\ast$-vectors are log-concave and unimodal.

Chvátal conjectured that every downset has a maximum-size intersecting family which is a star, that is, consists of all members of the downset containing a fixed element. Recently, Chang, Liu and Liu gave a proof of this conjecture, which follows as a corollary of more general results such as Kleitman's conjecture and a version of Kahn's conjecture (which they also prove). We give a short, direct (spectral) proof of Chvátal's conjecture, and a proof of Kleitman's conjecture along the same lines. We also prove a strengthening of Chvátal's conjecture concerning the projection packing number. Finally, we propose two spectral Chvátal conjectures which are motivated by extensive numerical experiments.

In 2006, Katzman showed that the first six Betti numbers of the Stanley–Reisner ring of a flag complex are field independent. He also found flag complexes on eleven vertices whose eighth Betti number depends on the field, and asked whether the seventh is always field independent. We answer this affirmatively by proving a stronger, purely topological result. Let $\tau(d)$ be the least number of vertices of a flag complex whose $d$-th reduced integral homology has torsion. We prove that $\tau(d)\geq d+10$ for every $d\ge0$. This bound yields the field independence of the seventh Betti number. Equivalently, combining our result with Katzman's, for every finite simple graph $G$, the first seven Betti numbers of the edge ideal $I(G)$ are field independent.

A sequence of integers $1=a_0<a_1<\cdots<a_k=n$ is called an addition chain of length $k$ if $a_j=a_s+a_t$ with $0\le s,t<j$ for all integers $j\in \{1,2,\ldots,k\}$. We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$. Here we investigate the distribution of the function $\ell$ through the counting function $$ F(m,r):=\#\{n\in [2^m,2^{m+1}):\ell(n)\le m+r\} $$ and show that, for every fixed $0<c<\log 2$, there exist positive constants $K_1$ and $K_2$ such that $$ K_1^r m^r\le F(m,r)\le K_2^r m^r $$ for all sufficiently large $m$ and all integers $m^{0.9}<r\le cm/\log m$. The upper bound also holds for every integer $r>m^{0.9}$. Moreover, denoting by $G(m,r)$ the number of distinct addition chains of length $m+r$ leading to an integer $n\in [2^m, 2^{m+1})$, we show that there exist positive constants $K_3$ and $K_4$ such that $$ K_3^r \left(\frac{m^2}{r}\right)^r\le G\left(m,r\right)\le K_4^r \left(\frac{m^2}{r} \right)^r $$ provided $m^{0.9}<r<m$. This improves and generalizes previous results on the minimal length of addition chains and addresses a question raised by Paul Erdős.

We introduce partition functors: algebraic structures indexed by partitions of finite sets and equipped with restriction and transfer maps along refinements. We construct a monad PD on the category of partition functors and on several categories of partition functors with additional multiplicative structure. For a partition ring, exponential elements in its associated completed ring of symmetric functions acquire canonical logarithms after applying PD. This gives rise to a universal exponential relation between multiplicative and additive power operations. For representation rings this can be used to recover the classical relation between symmetric powers and Adams operations, while for Morava E-theory it can be used to recover Ganter's exponential relation between symmetric powers and Hecke operators. We show that the representation rings of products of symmetric groups form the initial partition ring and that, for symmetric monoidal partition functors, the monad PD is closely related to symmetric invariant tensors and the divided power envelope. We also construct a symmetric monoidal partition ring carrying the universal exponential element, so that its image under PD carries the universal exponential relation.

We report an explicit construction of a 27 x 27 x 27 symmetric tensor with rational entries that has tensor rank 55 over the rational numbers and symmetric tensor rank 56 over the complex numbers, providing a small counterexample to Comon's conjecture over the rational, real, and complex numbers. The construction follows the framework of symmetric adjoins introduced by Shitov. The primary technical contribution of this work is to prove a special case of Conjecture 6 appearing in Shitov's seminal 2018 work.

We realize the $\mathrm{SU}(3)$ Verlinde sums $V_n(m)$, up to an explicit factor, as the values at $n$ of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on $m^2$ vertices. For fixed $m$, we express their generating function in terms of the logarithmic derivative of an associated even spectral polynomial and express this polynomial as an explicit iterated resultant. Exploiting permutation symmetry, we prove that this polynomial is a cube over $\mathbb{Q}$, apart from an explicit quadratic factor when $3\mid m$. This factorization yields shorter linear recurrences satisfied by $V_n(m)$ with constant coefficients. We also derive Binet-type formulas expressing $V_n(m)$ as finite linear combinations of powers of rescaled inverse squares of the roots of the spectral polynomial. These results provide an efficient algorithm for computing these Verlinde sums. Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.

We disprove the Reiner–Shimozono conjecture that products of key polynomials have nonnegative expansions in Demazure atoms. By relaxing the defining relations of Demazure modules, we obtain an exact coefficient-extraction formula that yields an explicit infinite family of counterexamples, including negative coefficients in twenty-eight variables. These examples answer van der Kallen's long-standing open question on relative Schubert filtrations negatively, already for tensor products of two dual Joseph modules. For individual atom coefficients, we give a sufficient criterion for positivity that is checkable in polynomial time in the binary input length, uniformly in rank. For each fixed rank $n\ge4$, this criterion recognizes a nonvanishing proportion of the positive atom coefficients as the bound on the composition entries tends to infinity.

In this paper, we establish formulas and sharp bounds for general Kulli-Basava indices and characterize graphs that attain these bounds. These indices have been shown to possess strong discriminating power for distinguishing nonisomorphic chemical structures. They are based on the edge neighborhood degrees of vertices in a graph. We also establish bounds for several classes of graphs, including triangle- and quadrangle-free graphs and graphs with a prescribed clique number. The formulas and bounds depend on the numbers of vertices and edges, the minimum and maximum edge neighborhood degrees, and the first Zagreb index.

A proof of the cycle double cover conjecture was recently announced, yielding an $8$-cycle double cover for every bridgeless graph. The stronger $5$-cycle double cover conjecture, which is still open, is equivalent to the statement that the edge set of every bridgeless claw-free cubic graph can be covered by at most four perfect matchings. Perfect matchings in bridgeless cubic graphs have been studied extensively, with two of the main conjectures in this area being the Berge–Fulkerson and the Fan–Raspaud conjectures. The latter, a consequence of the former, states that every bridgeless cubic graph admits three perfect matchings $M_1, M_2, M_3$ such that $M_1\cap M_2\cap M_3=\emptyset$. Here we show that the Fan–Raspaud conjecture is true for bridgeless claw-free cubic graphs. This also gives further information on the interaction of perfect matchings in a class where the $5$-cycle double cover conjecture requires control of four of them.

A graph is biplanar if it is the union of two planar graphs on the same vertex set. The largest chromatic number of a biplanar graph is known to lie between 9 and 12. The lower bound comes from Sulanke's graph, which has independence number 2, and a biplanar graph on 19 vertices with independence number 2 would have chromatic number at least 10. Gethner and Sulanke asked in 2009 whether such a graph exists. We show that it does not, and more generally that every biplanar graph with independence number at most 2 is 9-colorable. The proof embeds a hypothetical counterexample in the union of two sphere triangulations, enumerates with SAT modulo symmetries the 3271 graphs that pass a necessary filter for the complement of such a union, and shows with a SAT solver that none of them is such a complement; a matching argument reduces the general statement to this computation and one further case on 18 vertices. The computational part of the proof, including the completeness of the enumeration and every refutation, is checked in Lean 4, assuming three classical facts about planar graphs. The Lean development, the SAT instances, and the enumeration certificates are available on Zenodo.