Papers

Let $\TP_k$ denote the triangular pyramid with $k$ layers. For a graph family $\mathcal F$, let $\ex(n,\mathcal F)$ be the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal F$ as a subgraph. For a graph $H$, let $\cJ(H)$ be the family of subgraph-minimal graphs among all such graphs $H-I$, where $I\subseteq V(H)$ is an independent set. For every integer $k\ge4$ and all sufficiently large $n$, we prove that \[ \ex(n,\TP_k)=\max_{0\le m\le n}\{m(n-m)+\ex(m,\cJ(\TP_k))\}+O_k(n)=\left\lfloor n^2/4\right\rfloor+\Theta_k(n^{4/3}). \] Moreover, for $k=4$, we characterize all extremal graphs and show that \[ \ex(n,\TP_4)=\max_{0\le m\le n}\bigl\{m(n-m)+\ex\big(m,\cJ(\TP_4)\big)\bigr\} \] for sufficiently large $n$.

We introduce the leap group of a graph, the subgroup of the symmetric group on the vertex set generated by graph derangements. We summarize historical results on graph derangements, which we call hops, and prove additional existence and complexity results. The leap groups of various canonical graphs are computed, and their behavior under basic graph operations is analyzed. We offer partial results on which groups can appear as leap groups, and study hops and leaps for infinite and directed graphs.

The $\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_{p^s}^n$. A $\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_{p^s}$-additive code. A recursive construction gives such codes $H_p^{t_1,\dots,t_s}$, of type $(n;t_1,\dots,t_s)$ and length $p^t$, and a partial classification of them is known. In this paper, we complete the classification of these codes for every prime $p$. The main tool is the derived code, obtained by puncturing the code outside a block of coordinates determined by its kernel. For a nonlinear code $H_p^{t_1,\dots,t_s}$ with $t_1\geq2$, the derived code is a $\mathbb{Z}_{p^{s-1}}$-linear GH code of the same family, although the mechanism for odd $p$ differs from the binary one. As a consequence, for $t\geq3$, there are $\mathcal P(t+1)-\mathcal P(t)$ nonequivalent such codes of length $p^t$ if $p$ is odd, and $\mathcal P(t+1)-\mathcal P(t)-1$ if $p=2$, where $\mathcal P(n)$ is the number of partitions of $n$. In particular, this completes the classification of the $\mathbb{Z}_{2^s}$-linear Hadamard codes. We also prove a conjecture on the codes with $t_1\geq2$ and $t_s=0$.

Finite convexly ordered binary branching $C$-relations have been viewed as a notable exception to the broad applicability of the partite construction in structural Ramsey theory. The absence of a proof of their Ramsey property by the partite construction was noted by Bodirsky in his 2015 survey and pointed out again by Hubička and Konečný in their 2026 survey. We overcome this limitation by giving proofs based on the partite construction of the known Ramsey properties of the class $\mathcal{CT}$ of finite convexly ordered treeable meet semilattices and of the classes of finite convexly ordered $C$-relations with arbitrary branching and with binary branching. Our proofs do not rely on Milliken's tree theorem. The key is to expand $\mathcal{CT}$ by level predicates and a predecessor function so that the locally finite subclass method applies. We then transfer the resulting Ramsey witnesses to the corresponding reduct classes and to the classes of convexly ordered $C$-relations.

In 1994, Friedman, Joichi and Stanton conjectured that coefficients of the power-series $q+(1-q)\prod_{i=n}^{2n-1}\frac{1}{1-q^{i}}$ are nonnegative for $n\geq3$ and $n$ being odd. Let $\qbinom nm$ denote the Gaussian polynomial. In 2003, Prellberg and Stanton conjectured that coefficients of the power-series $(1-q)\qbinom nm\prod_{i=0}^{m-1}\frac{1}{1-q^{i+m}}$ are nonnegative for $0<m<n$, except when $n$ is even and $m\in\{2,n-2\}$. The latter conjecture implies the former. In this paper, we prove a stronger result concerning coefficientwise nonnegativity of another power series, which implies the conjecture of Prellberg and Stanton. The proof uses a Möbius correction from Gaussian polynomials, weight-preserving injections on partitions, explicit linearly independent vectors in the kernel of a derivation operator and the KOH identity.

In 1984, Bressoud and Subbarao established an interesting weighted partition identity connecting the partition function and the divisor function. Recently, the first author, jointly with Bhoria, Eyyunni, and Maji, extended the work of Bressoud and Subbarao and obtained the following analogous weighted partition identity. For any $n \in \mathbb{N}$, they obtained \begin{equation*} p^{(2)}(n) = \sum_{\pi \in \mathcal{D}(n)} (-1)^{ \# (\pi) }s(\pi) \left( \ell(\pi) - s(\pi) \right). \end{equation*} In the present paper, we provide a combinatorial proof of the above identity and extend the result for $p^{(k)}(n)$, where $p^{(k)}(n)$ denotes the number of partitions of $n$ into exactly $k$ distinct part sizes.

For a given graph $F$, a graph $G$ is $F$-irregular if all its vertices have pairwise distinct $F$-degrees, where the $F$-degree of a vertex $v$ is the number of subgraphs of $G$ containing $v$ and isomorphic to $F$. While the existence of $F$-irregular graphs has been actively studied for connected graphs $F$, the case where $F$ is disconnected remains entirely unexplored. In this paper, we provide a complete solution to this problem. We prove that for every disconnected graph $F$ with $|E(F)| \ge 2$, there exist infinitely many $F$-irregular graphs. Conversely, if $|E(F)| \le 1$, we show that no non-trivial $F$-irregular graph exists.

We construct trees with perfect matchings and independence sequences that fail log-concavity near the independence number $\alpha$. For every integer $k\geq1$, one family fails at $\alpha-k$ and has a witness of order $O(k^2\log(k+2))$. For every integer $t\geq2$, a single tree has simultaneous failures at $\alpha-1,\alpha-3,\ldots,\alpha-(2t-1)$, with a witness of order $O(t^3)$. All these trees satisfy $\alpha=N/2$, where $N$ is the number of vertices, and contradict Conjecture 4.2 of Ramos and Sun. A sufficient condition uses a fixed tree and the numbers of attached leaves. For fixed $k$ or $t$, explicit bounds give geometric decay of $a_j^2/(a_{j-1}a_{j+1})$ at these indices as the trees grow; $a_j$ counts independent $j$-sets. The proofs count independent sets by empty matching pairs. The Lean proof assistant checks formal proofs of the parameterized results and their quantitative bounds; exact 90- and 180-vertex examples illustrate the constructions.

In this paper, we study a partition function with parts separated by parity from a family of partition functions introduced by Andrews in 2019. We first show that its generating function admit representation in terms of mock modular forms of weight $3/2$. We then initiate a study of smallest-part partition function analogues of partitions with parts separated by parity, which, to the best of our knowledge, has not previously been investigated in the literature. We show that the generating functions of these smallest-part partition functions are likewise expressible in terms of mock modular forms, mock Maass theta functions and Zagier's $q$-series $H(q)$, which he describes as a “half-derivative of the Dedekind eta-function". Our study also leads to new combinatorial interpretations of the coefficients of the aforementioned mock modular form, function $\sigma^*(q)$, and mock theta functions $\psi(q)$ and $A(q)$ in terms of odd Ferrers graphs. Moreover, we provide a quantum $q$-series identity involving Ramanujan's famous $\sigma(q)$ function. We also derive the leading asymptotic term of one of these partition functions, whose generating function we first express in terms of $\sigma^*(q)$.

Goodman proved a lower bound on triangle counts which, in graphon language, states that every graphon of edge density $p$ has triangle density at least $2p^2-p$. Bennett, Dudek, Lidický, and Pikhurko established the corresponding bound for $C_5$, determining its minimum density at every Turán density $p=1-1/k$ with an integer $k\ge3$. We extend this bound to every odd cycle in the dense regime: for every odd integer $m\ge3$ and every graphon $W$ with edge density $p\ge2/3$, $t(C_m, W) \ge p^m-p(1-p)^{m-1}.$ For $2/3\le p<1$, equality holds precisely for balanced complete $k$-partite graphons with an integer $k\ge3$, up to graphon equivalence, and we establish qualitative stability at these densities. In the same density range, we also strengthen Kim and Lee's density-sensitive commonality inequality for odd cycles by adding the nonnegative term $t(P_{m-1},1-W)-(1-p)^{m-1}$ to the lower bound, where $P_{m-1}$ is the path with $m-1$ edges. To prove the main results, we develop a spectral excursion expansion and use Dirichlet averaging to reduce the resulting polynomial inequalities in arbitrarily many spectral variables to a one-variable problem. We prove the one-variable inequality through a uniform inequality comparing odd and even moments of shifted Gamma random variables. We also formalise the odd-cycle lower-bound theorem in Lean 4.

We introduce river ribbon tableaux, where a river of non-trivial ribbons is placed between two banks of single boxes. Compared to other ribbon peelings, this reduces the geometric differences across diagrams of different shapes, and the maximal number of ribbon configurations. This allows us to prove a uniform character bound for symmetric groups which, specialized to balanced diagrams, proves the conjecture of Moore and Russell. As an application, we prove a conjecture of Dousse and Féray on the number of standard tableaux of skew shapes.

Let $P_{n,k}$ count permutations of $[n]=\{1,\ldots,n\}$ whose longest increasing subsequence has length $k$. Chen conjectured that $(P_{n,k})_{k=1}^n$ is log-concave for every positive integer $n$. In this paper, we prove that Chen's conjecture is true when $n\ge 10^{10^{65}}$. The proof combines Gessel's determinantal formula with quantitative coefficient extraction, Tracy–Widom density estimates, and exact inequalities for the extreme lengths.

We study the local $\mathrm v$-numbers of binomial edge ideals from a combinatorial perspective. For each minimal prime $P_{\mathcal S}$ of $J_G$, indexed by a cutset $\mathcal S$ of $G$, we consider two combinatorial invariants, $\mathrm v^-_{\mathcal S}(G)$ and $\mathrm v^+_{\mathcal S}(G)$, defined in terms of transversals of suitable hypergraphs, and show that the local $\mathrm v$-number $\mathrm v_{P_{\mathcal S}}(J_G)$ satisfies $\mathrm v^-_{\mathcal S}(G)\leq \mathrm v_{P_{\mathcal S}}(J_G)\leq \mathrm v^+_{\mathcal S}(G)$. Under a suitable hypothesis on the connected components, we provide explicit graph-theoretic descriptions of these invariants and analyze when they coincide, giving a combinatorial way to determine the local $\mathrm v$-numbers. As an application, we derive a sufficient condition for this coincidence, which applies to a broad class of graphs. Furthermore, we identify a constant that controls the gap between $\mathrm v^-_{\mathcal S}(G)$ and $\mathrm v^+_{\mathcal S}(G)$, thereby showing that the local $\mathrm{v}$-number lies within a bounded interval. Applications to cycles and trees illustrate both the equality criterion and the resulting bounds.

A critical set in a Latin square is a set of entries that determines the square uniquely and is minimal with this property. We prove that every partial Latin square of order $n$ with fewer than $\lfloor n^2/4\rfloor$ filled cells has an even number of completions. Consequently every critical set in every Latin square of order $n$ has at least $\lfloor n^2/4\rfloor$ entries. Since the cyclic Latin square has a critical set of exactly this size, the minimum size of a critical set over all Latin squares of order $n$ is $\lfloor n^2/4\rfloor$. This confirms a conjecture attributed to Nelder (1979) and made independently by Mahmoodian and by Bate and van Rees. Hatami and Qian had shown that this minimum is at least $n^2/10^4$ for large $n$. The proof is a counting argument modulo two that uses only expansion of products, cancellation in pairs and the pigeonhole principle, and it is illustrated with partial Latin squares of orders three to seven. An appendix discloses the substantial use of AI in developing the proof and writing the exposition.

We study alternating Hamilton paths and cycles in the comparability graphs of finite posets. We prove that if a finite poset $P$ with at least two elements admits an alternating Hamilton path, then $P\times B_r$ admits an alternating Hamilton cycle for every $r\ge2$, where $B_r$ is the Boolean lattice. As a corollary, we settle a conjecture of Katona [Discrete Appl. Math., 2013]. Furthermore, we characterize the divisor lattices admitting alternating Hamilton paths and cycles.

We study non-additive rank-metric codes contained in the space of symmetric matrices over a finite field. Our main focus is on symmetric $2$-codes. For odd $q$, we introduce a quotient construction for $\mathrm{Sym}_3(q)$ in which rank-one differences are described by isotropic vectors of a nondegenerate quadratic form on $\mathbb F_q^4$. This leads to a correspondence with partial ovoids of the elliptic quadric $Q^-(5,q)$. Using a construction of Cossidente and Pavese, we obtain a non-additive symmetric $2$-code in $\mathrm{Sym}_3(q)$ of cardinality $q^4+2q^3$, improving the previously known general lower bound. We also give an explicit matrix description of this family. Lastly, we present a block-based construction for symmetric $2$-codes that extends the three-dimensional improvement to all odd dimensions. In particular, for every odd $n\geq3$ and every prime power $q>2$, we obtain \begin{equation*} A_q^sym(n,2) \ge q^{(n^2-1)/2}+q^{(n^2-3)/2}+q^{(n^2-9)/2}, \end{equation*} which is strictly larger than the maximum cardinality of an additive symmetric $2$-code with the same parameters. This yields the first infinite family (as the matrix dimension $n$ varies) of non-additive symmetric $2$-codes whose size is larger than the largest achievable size of an additive code with the same parameters; before this, such a gap had only been established for $n=3$.

Classical Turán theory relates the asymptotic extremal problem for an ordinary forbidden graph to its chromatic number. This principle fails when a forbidden copy must respect prescribed vertex colors, as occurs naturally in multipartite extremal problems and in embeddings with prescribed clusters. We study properly vertex-colored graphs whose hosts have a prescribed density between every pair of color classes, with containment interpreted through color-preserving homomorphisms. Our main theorem gives a finite directed reduction: for every fixed $m$ and $\varepsilon>0$, every sufficiently many-colored $d$-dense host admits a colored homomorphism from an $m$-layer half-blowup $D[m]$ of a finite digraph $D$ with density at least $d-\varepsilon$, where $D$ may be chosen with $\lceil\varepsilon^{-1}\rceil$ vertices. Consequently, the colored Turán density of every non-empty finite forbidden family equals the homomorphism Turán density of its associated acyclic orientations. We then determine the complete finite value set: these densities are exactly the Lagrangians of finite loopless digraphs, and hence are rational. The main technical input is a low-resolution sampling method that preserves all pairwise bipartite degree profiles on a large subfamily of color classes while keeping the sample size independent of the number of retained classes.

We give two counterexamples to Stapledon's Effectiveness Conjecture. Each counterexample is a $6$-dimensional lattice polytope with an order-two symmetry whose equivariant $H^*$-series $\varphi[t]$ is a polynomial with one coefficient that is not effective. For polytopes with a symmetry of order two, the goal of this paper is to build up a picture of the landscape of results for when the Effectiveness Conjecture holds. Our main result is a criterion, in terms of the faces of $P$, that determines when the Ehrhart ring of $P$ admits an invariant linear system of parameters of the correct representation type. We use these results to reduce the space of possible counterexamples to one of two cases. A randomised computer search produced counterexamples in each case. We also give an example where the equivariant $H^*$-series is effective but admits no invariant linear system of parameters, neither for $P$ nor any invariant subdivision of it, answering a question of Stapledon in the case of groups of order two.

We prove that $r(4,t) \ge \frac{t^3}{(\log t)^{2+o(1)}}$. This improves the lower bound of Mattheus and Verstraëte (Ann. of Math., 2024) by a factor of $(\log t)^{2-o(1)}$.

Extending the notions of Kneser graphs and Schrijver graphs, we define the $G$-stable Kneser graph $KG(G,k)$ (where $k\geq 2$ and $G$ is a graph on $n\geq 2k$ vertices) as the graph whose vertices are the $k$-element independent sets in $G$ and whose edges join pairs of disjoint independent sets. The well-known result of Lovász on the chromatic number of Kneser graphs implies that $\chi(KG(G,k)) \leq n-2k+2$ for any $G$. We call $G$ $k$-tight if equality holds, and we study the following question: Which graphs are $k$-tight? A result of Daneshpajouh et al.\ implies a characterization of $2$-tight graphs. We prove a necessary condition for a graph to be $k$-tight for $k\geq 3$ which extends this characterization, and we conjecture it to be sufficient. Another result gives a lower bound to the chromatic number of the $G$-stable Kneser graphs when $G$ is a disjoint union of cycles. We use this result to give a short proof of the fact that $4$-regular graphs on $3k$ vertices consisting of two vertex-disjoint cycles and $k$ cycles of length $3$ admit an independent set of size $k$.

For a graph $H$, the suspension $\widehat H$ of $H$ is a graph obtained from $H$ by joining a new vertex to every vertex of $H$. The generalized Turán number $\textrm{ex}(n,K_3,\widehat H)$ is the maximum number of triangles in an $n$-vertex $\widehat H$-free graph. For fixed $k \ge 4$, Mubayi and Mukherjee conjectured that \[ \textrm{ex}(n,K_3,\widehat P_k) =\left\lfloor\frac{k-2}{2}\right\rfloor\frac{n^2}{8}+o(n^2), \] as $n \to \infty$, where $P_k$ is the path on $k$ vertices. They verified the conjecture for $k=4,5,6$. In this paper, we determine the asymptotic value of $\textrm{ex}(n,K_3,\widehat P_k)$ for every $k\ge 4$ via a unified approach and thus prove the conjecture of Mubayi and Mukherjee.

For a Bruhat interval $[x,y]$ in an arbitrary Coxeter group which is a lattice, we give a type-uniform construction of a convex polytope whose complete face lattice realizes $[x,y]$. This gives a polytopal realization of the coincidence between classical Kazhdan–Lusztig polynomials for lattice Bruhat intervals and Stanley's toric $g$-polynomials. Consequently, the Kazhdan–Lusztig polynomial of a lattice Bruhat interval cannot have internal zeros, satisfies a multiplicative strengthening of monotonicity, and obeys the generalized upper bound theorem for polytopes. For lattice Bruhat intervals in finite Weyl groups, we explain the relation with the toric geometry of Richardson varieties and derive formulas for their local and global intersection cohomology, as well as a Boolean criterion for rational smoothness. We conjecture that the coefficients of these Kazhdan–Lusztig polynomials form an $M$-sequence.

We completely characterize the solvability profiles of tasks for two autonomous mobile robots in the plane under variable disorientation, chirality, and rigid movements. The four standard capability models and the schedulers considered here have a known quotient order Q of thirteen computational classes. A task's solvability profile is the set of classes in which it is solvable, and is necessarily an upset of Q. We prove that every upset is realizable and construct a minimum exact disjunctive basis of thirteen concrete tasks. The basis contains one task for each principal upset. For every antichain of Q, the disjunction of the corresponding tasks has profile exactly equal to the union of their profiles. This exactness is not implied by the individual profiles: an algorithm may satisfy different disjuncts in different executions. We establish it by constructing, for every algorithm in each maximal class outside the target union, a single admissible fair execution violating all selected predicates simultaneously. Consequently, the realizable profiles form the entire finite distributive lattice Up(Q): there are exactly 54 profiles, counted by the antichain polynomial 1 + 13z + 24z^2 + 16z^3, and each has a canonical disjunctive representation using at most three principal tasks. We also identify six tasks whose solvability signatures recover every inclusion and incomparability in Q, and prove that six is minimum even among arbitrary separating tasks. Their profiles are precisely the principal upsets generated by the join-irreducible elements of Q. In contrast, any task family generating every profile by exact disjunctions must contain all thirteen principal profiles. Thus our exact disjunctive basis has minimum size and is unique at the profile level.

A complete mapping of a finite group $G$ is a permutation $\theta$ of $G$ such that $x\mapsto x\theta(x)$ is also a permutation of $G$. Complete mappings have applications in cryptography, particularly in the design of round functions for cryptographic systems. The study of their parity has led to the question of which finite groups admit complete mappings of both parities. A useful approach to this question is to construct complete mappings stabilizing suitable subgroups, in particular Sylow $2$-subgroups. We show that, for every $n\geq2$, every prime power $q$, and every $Z\leq Z(\mathrm{SL}_n(q))$, the group $\mathrm{SL}_n(q)/Z$ admits a complete mapping stabilizing a Sylow $2$-subgroup if and only if $(n,q)\neq(2,2)$.

After a basketball game, each of the $n$ players on Team A must shake hands with each of the $n$ players on Team B. What is the minimum number of (reversible, ambidextrous) gloves required so that zero germ transmission occurs? We show that the answer is $3n/2+o(n)$ and develop a general theory for prescribed handshake sets.

When characterizing graphs defined by forbidden induced subgraphs, it is often desirable to understand how a graph $G$ can be generated incrementally from a given induced subgraph $H$. In this paper, we study a number of problems within this framework. First, for several notions of connectivity, we determine all minimal induced subgraphs of $G$ that contain $H$ and attain the connectivity of $G$. Second, we establish how to grow $H$ inside $G$ while preserving twin-freeness and certain forms of connectivity. Third, we establish finite test theorems for various graph classes, proving that agreement with the recursively generated graphs for a few initial iterations forces agreement at all subsequent iterations. Consequently, such an infinite graph family can be completely determined by finitely many of its initial generations. As an application, we characterize $\{\mathbb L_1,\mathbb L_3\}$-free graphs, where $\mathbb L_1$ is the claw and $\mathbb L_3$ is another forbidden graph for line graphs. Our structural results reduce this problem to a finite verification, which is completed through computer-assisted computation.

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), its $\ell$-degree Turán density $\gamma^{(k)}_\ell(\mathcal{F})$ is the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all integers $k>\ell>2$ and every $\alpha\in[0,1)$, there exists an infinite family $\mathcal{F}$ of $k$-graphs such that $\gamma^{(k)}_\ell(\mathcal{F})=\alpha$, while every finite subfamily $\mathcal{F}'\subset \mathcal{F}$ satisfies $\gamma^{(k)}_\ell(\mathcal{F}')>\alpha$. This confirms the conjecture of Mubayi and Zhao on codegree Turán densities for every $k > 3$. As an immediate consequence, the set $\Gamma^{k,\ell}_{\infty}$ of $\ell$-degree Turán densities of nonempty $k$-graph families is $[0,1)$ whenever $k>\ell>2$, which answers a question of Lo and Markström from 2014.

A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class are linearly ordered by inclusion. We study the generalized spectral determination of chain graphs of adjacency rank $4$. We first show that generalized cospectral chain graphs have the same number of isolated vertices, reducing the problem to the connected case. For connected rank-$4$ chain graphs, we derive four explicit equations in the cell parameters that are necessary and sufficient for generalized cospectrality. These equations reduce to a quadratic equation in a single parameter. As a consequence, every connected rank-$4$ chain graph has at most one generalized cospectral mate, and the possible mate can be reconstructed explicitly. We further obtain a simple sufficient condition for a rank-$4$ chain graph to be determined by its generalized spectrum (DGS) and, in particular, prove that every double star graph is DGS. Finally, we construct an explicit infinite family of non-double-star rank-$4$ chain graphs that are DGS but not determined by their adjacency spectra, with both the graphs and their complements being connected.

A tensor product is known for commutative Rota–Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product $A\otimes B$ and can be substantially larger. We show that, after passing to augmented Rota–Baxter algebras, the ordinary tensor product itself carries a natural Rota–Baxter operator which makes the category monoidal with the usual associator and unit constraints. The resulting structure is intrinsically nonsymmetric: over a field of characteristic zero, this operator and its flip-conjugated counterpart are the only possibilities among linear combinations of five natural terms, while over any field the full category admits no braiding. Its comonoid objects are precisely Rota–Baxter cocycle bialgebras. Moreover, the free Rota–Baxter adjunction yields a bimonad and free cocycle bialgebras on ordinary bialgebras, with compatible Hopf structures for connected filtered inputs, in both the associative and commutative settings.

Let $G$ be a thrackle, where a thrackle is a graph drawing in which each edge meets every other edge exactly once, either at a vertex or at a crossing. It is conjectured that, for all thrackles, the number of edges is less than or equal to the number of vertices. Fulek and Pach created a program to determine whether or not a graph can be a thrackle. Through a reconstruction and improvement of their algorithm, we computationally prove a key dumbbell to be non-thrackleable. Using this exclusion, we improve Xu's 8-face half-edge bound from five to four. Finally, using this stronger restriction with Xu's framework, together with separate arguments for graphs containing triangles and disconnected graphs, we improve the known upper bound toward this conjecture to $m\le1.385(n-1)$ for $n\ge4$.