Papers

For fixed $k\ge3$, we determine the critical law of totally isotropic $r$-spaces for a uniform random map $\Theta_N:\Lambda^k\mathbb {F}_q^N\to\mathbb {F}_q^m$. At the exact balance $m\binom rk=r(N-r)$, the entire null configuration is asymptotically an independent Bernoulli subset of $\operatorname{Gr}(r,\mathbb {F}_q^N)$ in total variation, uniformly in $q$ and $m$. Consequently, the counting measure $\Xi_r$ is asymptotically a Poisson point process, and its total mass $X_{N,r}$ is asymptotically Poisson. An exterior-rank stability theorem shows that near-extremal families decompose into Grassmann clusters with uniformly controlled span deficiency. We obtain quantitative rates and identify the first exterior-dependence scale. We also show that rare null $(r+1)$-spaces force high-order factorial-moment divergence, while the fixed-target bilinear cases $m=1,2$ exhibit non-Poisson critical behavior.

Let $q$ be odd. The trace conic in $\mathbb{F}_{q^3}$ determines a Singer difference set in $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^\times$ and a natural square-class lift to $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^{\times 2}$. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by $2\sqrt{q}$. The proof improves the naive six-puncture Weil bound by exploiting a projective Klein-four symmetry of the associated rank-one local system. The resulting nontrivial cocycle produces a quaternionic action on its four-dimensional cohomology, while Frobenius symmetry reduces the relevant trace to two Weil-scale eigenvalues. As an application, the oriented conic yields an explicit Singer-invariant signing of the point-line incidence graph of $\mathrm{PG}(2, q)$. The corresponding dihedral Cayley graph is a connected Ramanujan double cover. Thus a conic lift already known in finite-geometric constructions has an additional Ramanujan spectral property governed by its odd multiplicative character sums.

We prove that every $4$-partite $4$-uniform hypergraph $H$ with matching number $\nu(H)=2$ satisfies $\tau(H)\le 6$, where $\tau$ denotes the vertex-cover number. This confirms a claim made by Tuza in his 1979 manuscript but never published with a proof, and closes the case $(r,\nu)=(4,2)$ of Ryser's conjecture. The best previous bound was $\tau\le 7$, an integrality consequence of the theorem of Haxell and Scott (2012). The proof uses Gyárfás's intersecting-case theorem ($\tau\le 3$ for intersecting $4$-partite $4$-uniform families), a short projection lemma (four base-disjoint edges in an intersecting family force a two-element cover), and Kőnig's matching theorem.

For each fixed integer $K\ge2$, we consider the Enots–Wolley sequence in which every term after the initial $1,2$ is required to have between two and $K$ distinct prime divisors. We prove that every integer satisfying this restriction occurs. If an exact prime support $T$ were selected only finitely often, then after a finite cutoff the terms meeting $T$ would form short episodes, and every full term would force an earlier proper term at comparable numerical height. The case $|T|=K$ is then ruled out directly. In the remaining case $|T|<K$, at a record proper value $H$, greediness forces every unblocked rank-$K$ integer below $H$ containing exactly one prime of $T$ to have occurred earlier. Fixed-order Landau estimates show that this proper population has order $H(\log\log H)^{K-2}/\log H$, while the entire possible full population on the same scale has strictly smaller logarithmic order. This contradiction proves surjectivity. The theorem concerns each fixed rank cap and does not settle surjectivity of the unrestricted Enots–Wolley sequence.

We resolve a conjecture on the law of large numbers for coherent measures on Young diagrams associated with Macdonald-nonnegative specializations of the algebra of symmetric functions for all $0\le q,t<1$. We prove that row and column lengths divided by the diagram size converge to the corresponding specialization parameters. The proof combines a monomial estimate for skew Macdonald polynomials with exponential tilting and binomial decompositions obtained from the branching rule for the polynomials.

We present a procedure that, given a pair of coprime integers, produces a one-vertex triangulation of the three-sphere containing the corresponding torus knot as an edge. The construction is canonical, and produces a three-sphere triangulation from a pair of minimal layered triangulations of the solid torus. The size of the triangulation is, in the worst case, linear, but can be as small as logarithmic in the crossing number of the knot, motivating this triangulation size as an organising principle for knots, alternative to the crossing number.

We give eight-vertex counterexamples to the equality characterization in Conjecture 20 of Chen, Guo, Li and Wang, Electron. J. Combin. 32(4) (2025), P4.37. The examples occur at the included endpoint $k=3n/4$ and extend to infinite families, including a family for which both the graph and its complement are connected. These examples do not contradict the numerical inequality in that conjecture.

Single-machine asymmetric weighted earliness–tardiness (AWET) scheduling asks how to sequence jobs around a common synchronization date when early and late completion incur unrelated job-dependent penalties. At the boundary nonrestrictive date $d=\sum_jp_j$, a compact V-shaped schedule reduces the continuous-time problem to a quadratic choice of a nonempty early set. We establish four complementary results for this model. First, the positive-integer problem is strongly NP-complete by a unary-polynomial reduction from Restricted Exact Cover by 3-Sets. Second, unrestricted AWET admits a polynomial-time $(3+2\sqrt2+\varepsilon)$-approximation based on an anchored semidefinite relaxation and deterministic marginal thresholding. Third, when the earliness and tardiness ratio orders are strict reversals, the problem is weakly NP-complete but has an exact two-resource pseudopolynomial dynamic program. Fourth, for fixed total refinements whose ratio permutation is separable, an exact separating-tree recurrence and coordinated geometric trimming yield an FPTAS. The proofs use different manifestations of the same canonical objective: scale-separated prefix penalties, positive-semidefinite minimum-kernel covariance, a dominant completed load square, and a bounded four-coordinate decomposition interface. Together, the results show that the decisive issue is not merely whether the two ratio orders agree, but whether their interaction can be controlled by a global certificate or compressed into a bounded constructive interface.

We give a polynomial-size linear programming bound for $q$-ary all-symbol locally recoverable codes with locality parameters $(r,\delta)$, without assuming linearity. The key idea is to keep, for every ordered pair of codewords and every selected recovery view, the joint Hamming weight on the helper set, the recovered coordinate, and the rest of the code – rather than collapsing this triple into a single distance, as earlier formulations do. Averaging this three-block distribution over recovery views of the same length yields exact identities linking it to the global distance distribution, together with nonnegative product-Krawtchouk constraints that encode locality and spectral positivity simultaneously. The resulting LP has polynomially many variables, its optimum dominates the ordinary Delsarte bound, and an earlier outside-distance formulation, the convex-hull bound of Li–Wei–Xiong, and the dual-based bound of Gruica–Jany–Ravagnani all arise from it as coarser marginals. Exact rational certificates over $q=2,3,4$ show the bound is strictly stronger than the best of these prior LPs in thirteen of fifteen tested cases, pinning down seven exact maximum code sizes and twelve exact maximum linear dimensions.

For a finite set $S$ with $\lvert S\rvert = N$, the number of families $\mathcal{B} \subseteq \mathcal{P}(S)$ that are topological bases is $\#(N) = \sum_{\mathcal{T} \in \operatorname{Top}(S)} 2^{\lvert\mathcal{T}\rvert - \lvert\mathcal{M}_{\mathcal{T}}\rvert}$, where $\mathcal{M}_{\mathcal{T}}$ is the canonical minimal basis of minimal open neighborhoods. The identity is proved in Lean 4 / Mathlib (`CARDB.lean`): bases generating $\mathcal{T}$ are exactly the sets with $\mathcal{M}_{\mathcal{T}} \subseteq \mathcal{B} \subseteq \mathcal{T}$. The small-$N$ table and the discrete-dominance sandwich are proved in `CARDB/SmallN.lean` and `CARDB/Asymptotics.lean`.

The deformed \(q\)-exponential function \[ e_q(z,u) = \sum_{n=0}^{\infty} u^{\binom{n}{2}} \frac{z^n}{(q;q)_n} \] provides a common framework containing several classical \(q\)-exponential functions as particular cases, including the Jackson \(q\)-exponentials \(e_q(z)\) and \(E_q(z)\). In this paper we investigate the multiplicative inversion problem \[ e_q(z,u)e_q(-z,v)=1, \] and determine all pairs of deformation parameters \((u,v)\) for which this identity holds. To this end, we introduce a family of coefficient polynomials whose common zeros characterize the inversion property. A geometric analysis of the first nontrivial coefficients reduces the problem to two parameter branches. The symmetric branch is excluded through a parity phenomenon, while the nonsymmetric branch is completely determined by the first two coefficient constraints. As a consequence, we prove a rigidity theorem showing that \[ e_q(z,u)e_q(-z,v)=1 \] if and only if \[ (u,v)=(1,q) \qquad\text{or}\qquad (u,v)=(q,1). \] Thus the classical Jackson inversion identity is rigid within the deformed family and no new multiplicative inversion identities arise from the deformation parameter.

For a crystallographic root system ${\mathfrak D}$ we consider the simplicial complex $\Delta_{\mathfrak D}$ of all antichains in the root poset of ${\mathfrak D}$. We show that $\Delta_{\mathfrak D}$ is shellable if and only if ${\mathfrak D}$ is $A_n$, $B_n$, $D_3$ or $G_2$. Since antichains in types $A_n$ and $B_n$ can be identified with Dyck paths and symmetric Dyck paths, respectively, this yields a simplicial complex on Dyck paths. Indeed, in type $A_n$, shellability can be extended to rational Dyck paths. The $f$- and $h$-triangles then yield statistics on (symmetric/rational) Dyck paths. We determine these statistics for $A_n$ and $B_n$ and leave the case of rational Dyck paths as an open problem.

Coven and Meyerowitz gave two cyclotomic conditions, (T1) and (T2), which characterize integer tiles whose cardinalities have at most two distinct prime factors. We prove that the same characterization holds without this restriction. The proof uses a reduction in Chinese remainder coordinates: slicing by the lowest base-p digit produces sets with a common tiling complement in a group of order smaller by a factor of p. This reduction preserves the cyclotomic data needed for an induction on the exponents in (T2).

Consider the linear form $L(x,y,z,w) = 3x+y-2z-2w$. For a positive integer $N$, denote by $r_L(N)$ the largest size of a subset of $\{1,2,\dots,N\}$ that avoids nontrivial solutions to $L = 0$. We show that $r_L(N) = \Omega(N^{0.5608687})$, improving the lower bound for Problem 16 in Green's list of open problems. Our proof uses the method of combinatorial degenerations to turn a finite certificate into large solution-free sets. In fact, we can improve Ruzsa's lower bound of $N^{1/2-o(1)}$ for many four-variable equations. Consider $L(x,y,z,w) = ax+by-cz-dw$ with $a,b,c,d\in \mathbb Z_{>0}$, $a+b=c+d$, $\{a,b\} \neq \{c,d\}$ and $abcd$ not a square. We show that there is $\varepsilon_L > 0$ such that $r_L(N) = \Omega_L(N^{1/2 + \varepsilon_L})$. Furthermore, we show that for primitive translation-invariant linear forms $L$ in $s$ variables, the lower bound $\Omega_s(N^{1/(s-1)})$ coming from a greedy construction is never optimal.

In this paper, we settle a conjecture of Polujan and Pott by constructing an explicit, infinite family of Maiorana–McFarland bent functions $f_t$ in $2(2^t-1)$ variables with algebraic degree $\deg(f_t) = t + 1$ for any integer $t \ge 2$. Our construction builds upon a minimal commutative algebra $I_t$, which naturally induces a triangular set-partition polynomial permutation $P_t$. By identifying an elementary abelian subgroup within the direct sum $ I_t \oplus I_t^*$, we establish an explicit nonlinear coordinate transformation that pulls $f_t$ back to a canonical quadratic form. This linearizes the translation development $\operatorname{Dev}(D_{f_t})$ under an exotic group structure and proves that it is isomorphic to the classical symplectic design $S^\pm(2(2^t-1))$, thereby fully resolving the conjecture.

Orbit-span and dimension problems for designs have been studied in several classical association schemes using a variety of methods. Recently, through a detailed analysis of total trades, Ghorbani et al. showed that the orbit of a fixed combinatorial design asymptotically attains the full dimension permitted by the design equations. Analogous dimension-saturation results for the global spans of index-one designs in the bilinear forms and Grassmann schemes were obtained via laborious eigenvalue computations. In this paper, we work with the top fiber $X$ of a graded poset carrying a compatible transitive action of a finite group $G$, and assume that the induced Schurian association scheme on $X$ is commutative. For the multiplicity-free decomposition $\mathbb{C}^{X}=U_{0}\oplus U_{1}\oplus\cdots\oplus U_{m}$, we prove that, under explicit spectral and quantitative conditions, the $G$-orbit of the characteristic vector of any $t$-design spans the maximal submodule $U_{0}\oplus U_{t+1}\oplus U_{t+2}\oplus\cdots\oplus U_{m}$ allowed by the $t$-design definition. We verify these conditions asymptotically for the Hamming, bilinear forms, Johnson, and Grassmann schemes. This recovers the fixed-orbit saturation theorem of Ghorbani et al. for combinatorial designs, gives a new result for orthogonal arrays, and asymptotically extends the previous bilinear forms and Grassmann results to orbit spans of individual designs of arbitrary fixed index.

The Interconnectedness Coefficient (IC) is a bounded semi-local graph-theoretic node measure designed to identify connector vertices between cohesive network regions. Such connector vertices, also referred to as bridging nodes, may mediate between locally cohesive regions even when they are neither hubs nor themselves highly clustered. The IC preferentially assigns high values to weakly clustered focal vertices whose adjacent vertices remain strongly clustered after exclusion of the focal connection. Candidate vertices are required to have degree at least two. The construction is partition-free, uses information within radius two, and requires no predefined community or module partition. The range and extremal properties of the score are derived analytically. Exact graph families isolate its maximal response to fully cohesive branches, its controlled response to a single cohesion defect, its invariance under a cohesion-free hub extension, and a sharp fragmentation threshold. A separate application to a Human Interactome Map reveals pronounced degree-dependent stabilization of IC values near the network's mean clustering level. This behavior follows directly from the multiplicative definition. If focal clustering tends to zero while mean leave-one-out cohesion in the neighborhood stabilizes, the IC converges to that neighborhood-cohesion level. Among the highly ranked IC vertices are proteins with established interface, scaffold, and adaptor roles in molecular complexes. The IC is therefore positioned as a semi-local connector measure for cohesive network regions.

Let $G$ be a finite group with $t$ conjugacy classes, and let $S_N$ be the symmetric group. Let $Z_t(N)$ be the number of zeros in the character table of the wreath product $G\wr S_N$. We prove \begin{equation*} Z_t(N)\ge \frac{2p_t(N)^{2}}{\log \frac{N}{t}}\left(1+O\left(\frac{\log \log \frac{N}{t}}{\log \frac{N}{t}}\right)\right), \end{equation*} where $p_t(N)$ is the number of $t$-multipartitions of $N$.

We prove that for additive complements $A$ and $B$, if $$ \limsup_{x\to+\infty}\frac{A(x)B(x)}{x}<\frac43, $$ then $$ A(x)B(x)-x\rightarrow+\infty\qquad(x\to+\infty). $$ This improves the earlier upper bound $3-\sqrt{3}$ due to Fang and Chen.

We prove a quantitative polynomial Roth theorem for corners in \(\mathbb F_p^2\) for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials \(\phi_1,\phi_2\in \mathbb \mathbb{F}_p [y]\) are of degree $\leq d$ vanishing at $0$ and are not linearly dependent, then every \(A\subset\mathbb F_p^2\) with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner $$ (x_1,x_2),\qquad (x_1+\phi_1(y),x_2),\qquad (x_1,x_2+\phi_2(y)) $$ for some \(y\in\mathbb F_p^\times\). This improves the estimate $p^{2-1/16}$ of Han–Lacey–Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han–Lacey–Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map \(\widetilde F:W\to\mathbb A^3\) from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin–Schreier sheaf and Katz–Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An \(\ell^2\) matrix estimate adapted to such sets completes the resonant case.

Let $T_{m,n}$ be the tadpole graph obtained by joining a cycle $C_m$ to a path $P_n$ by a bridge. We prove that the independence polynomial of every tadpole graph is unimodal and establish sharp bounds for its mode. The unimodality result follows from a general criterion for graphs obtained by attaching a path to a fixed vertex. Over a field of characteristic zero, we also give a complete classification of the pairs $(m,n)$ for which the Artinian algebra defined by the edge ideal of $T_{m,n}$ together with the squares of all variables has the weak Lefschetz property.

For \(n\ge2\), let \(A(n)=\pi(n)-\omega(n)\) and \(B(n)=\phi(n)-\pi(n)+\omega(n)\) denote the numbers of prime and nonprime totatives of \(n\), respectively, with \(1\) included in \(B(n)\). We construct an explicit injectively parametrized family \(\mathcal C_n\) of composite totatives by completing suitably restricted squarefree products of prime totatives with a larger prime totative that is the unique largest prime factor, making the parametrization recoverable. We prove \(\lvert\mathcal C_n\rvert/A(n)\ge (e^{-\gamma}-o(1))\log A(n)/\log\log A(n)\), realizing the sharp classical leading constant \(e^{-\gamma}\). For the same family, with the same defining parameters, we obtain an effective refinement with error of order \((\log\log A(n))^{-1/4}\) and absolute effectively computable constants. Classical minimal-order results and the prime number theorem show that \(e^{-\gamma}\) is optimal: no family contained in the nonprime totatives can satisfy a uniform lower bound at this scale with a larger leading constant. We also determine exact eventual thresholds. Let \(N_k\) and \(M_k\) be the least integers such that \(\phi(n)>k\pi(n)\) for every \(n\ge N_k\) and \(B(n)>kA(n)\) for every \(n\ge M_k\), respectively. We prove \(M_k\le N_{k+1}\) for every \(k\ge1\), determine \(N_1,\ldots,N_6\) and \(M_1,\ldots,M_5\) exactly, and obtain \(M_k=N_{k+1}\) for \(1\le k\le5\).

Let $X_w$ be a Type A local complete intersection Schubert variety. We prove Contact Rigidity: the existence of two smooth singular components forces some pair of singular components to contain a common Schubert subvariety of codimension one in each. If the singular locus is a single smooth component $X_z$, the rational comparison kernel is $IC_z$, and $P_{u,w}(q)=1+q^{(\ell(w)-\ell(z)-1)/2}$ for every $u\leq z$. The first proof combines pattern avoidance with a computer-assisted finite overlap classification, rectangle inheritance, and extremal repairs. The second establishes the hypotheses of Woo's theorem and uses Euler characteristics and Bruhat triangularity to identify the entire perverse kernel.

Let $\nu(G)$ be the maximum number of edge-disjoint triangles in a graph $G$ and $\tau(G)$ the minimum number of edges meeting every triangle. Tuza conjectured that $\tau(G)\le 2\nu(G)$. We prove that $\tau(G)\le (165/59)\nu(G)$. The constant $165/59\approx 2.797$ improves the bound $66/23\approx 2.870$ that Haxell proved in 1999. The key observation is that, for a suitable red-blue coloring, the families left over in Haxell's construction contain every triangle with exactly one red edge. Such a family $\mathcal{F}$ admits an exchange that forces certain red edges to lie in a single triangle once the blue edges of a maximum packing are deleted, which gives $\tau(\mathcal{F})\le (8/3)\nu(\mathcal{F})$.

We show that if an n-vertex 5-connected graph has at least 4n-10 edges, then for any choice of five of its vertices, we can contract disjoint connected subgraphs containing these vertices to obtain $K_5$ as a minor. The bound on the number of edges is the best possible.

Let $G=(V,E)$ be a graph. The eccentric connectivity index of $G$ is defined as $$\xi^C(G)=\sum_{u\in V(G)}d_uecc(u)$$ where $d_u$ and $ecc(u)$ are the degree and eccentricity of $u$, respectively. For a strongly connected digraph $D=(V,A)$, the eccentric connectivity index is defined as $$\xi^C(D)=\frac{1}{2}\sum_{u\in V(D)}(d_u^++d_u^-)mecc(u)$$ where $d_u^+$ and $d_u^-$ are the out-degree and in-degree of $u$, respectively, and $mecc(u)$ is its m-eccentricity with respect to the maximum distance $md(u,v)=\max\{\vec d(u,v),\vec d(v,u)\}$. In this article, give formulas and bounds for the eccentric connectivity index of Cartesian and strong products of strongly connected digraphs and discuss the corresponding equality cases. Also, an attempt is made to study the self-centeredness of these products and establish conditions under which the Cartesian and strong products are self-centered.The results are extended to products of several digraphs.

A boxed mesh pattern is a mesh pattern whose selected entries lie in an empty axis-parallel rectangle. We classify coincidences of boxed patterns with classical and vincular patterns, exhibit a genuinely bivincular coincidence, and prove that no boxed–bivincular coincidence occurs for patterns of length at least five. Together with known results, this shows that every boxed pattern of length at least five has factorial growth and hence fails the Stanley–Wilf property. At length four, one exceptional orbit is enumerated by the semi-Baxter numbers, while the remaining exceptional orbit, $\{2143,3412\}$, is unresolved; we conjecture that it has factorial growth. For Box(123), we derive an exact maximum-insertion identity and prove the subfactorial upper bound $2^{5n}n^{\beta n}$, where $\beta=\log_2(2\cos(\pi/7))<0.85$. A closed enumeration remains open. We also prove a general first-moment formula for boxed mesh patterns that depends only on the length of the underlying pattern; in particular, the expected number of Box(123) occurrences in a uniformly random permutation of length $n$ is asymptotic to $n\log n$. For Box(12), we identify the occurrence statistic with the up-degree in the strong Bruhat order, obtaining its maximum, its mean, and an exact insertion identity for the distribution polynomials. We conjecture that the coefficients of these polynomials are unimodal.

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

We say that $v\in V(G)$ is a good vertex if the Kirchhoff index remains unchanged when $v$ is removed, i.e. $Kf(G)=Kf(G-v)$. In 1991, Šoltés studied the Wiener index of a graph and posed the problem of identifying graphs for which the removal of an arbitrary vertex preserves the Wiener index. In this paper, we explore a similar concept: identifying Kirchhoff Šoltés graphs, i.e. graphs in which all vertices are good vertices. We show that the cycle $C_5$ is a Kirchhoff Šoltés graph. We consider several relaxed versions of the Kirchhoff Šoltés problem, where the primary objective is to identify graphs containing at least one good vertex. One of them is the \textit{$\beta$-Kirchhoff Šoltés problem}, which seeks to find an infinite family of graphs in which the proportion of good vertices is at least $\beta$, with $\beta \in (0,1]$ being a specified rational number. Another one involves constructing infinite families of graphs where the proportion of good vertices increases and asymptotically approaches a given real number $\gamma\in (0,1]$ as the order of the graph grows. We demonstrate that both relaxed versions have infinitely many solutions. In particular, we prove the existence of infinitely many graphs for which the proportion $\beta$ of good vertices, $1/7\leq \beta<1/5$ tends to a certain irrational number. Furthermore, we prove the existence of infinitely many graphs with half good vertices, and for each $s\in\mathbb{N}$, we construct an infinite family of graphs whose proportion of good vertices tends to $\frac{s+1}{2s+1}$. These findings could be pivotal in addressing the original problem of determining whether there are additional solutions beyond $C_5$.

Let $A$ be a zero-dimensional monomial algebra over an algebraically closed field of characteristic zero, that is, a finite-dimensional quotient of a polynomial ring by a monomial ideal. Its automorphism group $G$ is a linear algebraic group, described through the homogeneous nilpotent derivations of $A$. We analyze the structure of $G$ in detail. Its identity component $G^0$ is a semidirect product of its unipotent radical and a reductive subgroup isomorphic to a product of general linear groups, and for each root degree we characterize when the associated derivations give rise to an additive root subgroup, and determine its dimension. Using the Lie brackets of these derivations, we then give an explicit algorithm that produces, out of the minimal monomial generators of the ideal, a family of root subgroups generating $G^0$ together with a maximal torus. Such a family is minimal in the generic case. We also show that the component group $G/G^0$ can be arbitrary: every finite group arises as the component group of the automorphism group of some zero-dimensional monomial algebra. Finally, we apply these results to the algebras $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$, showing that the subgroup generated by a maximal torus and the outer root subgroups is exactly the subgroup of automorphisms with constant Jacobian determinant, and we deduce from this a new proof of Anick's theorem on the density of the tame automorphisms of $\mathbf{k}[\mathbf{x}]$.