Papers

A graph $G$ is $\tau_k$-maximal if $G$ contains no subgraph admitting $k+1$ edge-disjoint spanning trees, while the addition of any edge in the complement of $G$ yields a subgraph that admits $k+1$ edge-disjoint spanning trees. In this paper, we prove that for any integers $k\geq 1$ and $n\geq 2k+2$, every $\tau_k$-maximal graph of order $n$ satisfies $|E(G)|\leq (k+1)(n-1)-1$. Furthermore, we construct a family of $\tau_k$-maximal graphs on $n\ge 2k+2$ vertices that have exactly $(k+1)(n-1)-1$ edges, which establishes the tightness of the upper bound. Then we conjecture that every $\tau_k$-maximal graph on $n$ vertices has exactly $(k+1)(n-1)-1$ edges, and we verify the conjecture for the case $k=1$.

The largest matching root $\mu(G)$ of a graph $G$ is that of its matching polynomial. In this paper, all limit points of the largest matching roots of graphs are determined. More precisely, we identify the limit points of the largest matching roots of graphs less than $\tau^{\frac{1}{2}}+\tau^{-\frac{1}{2}}$. For any $\gamma \geq \tau^{\frac{1}{2}}+\tau^{-\frac{1}{2}}$ with $\tau=\frac{\sqrt{5}+1}{2}$, there exists a graph sequence $\{G_i\, |\, i\in \mathbb{N}\}$ such that $\lim\limits_{i \rightarrow \infty}\mu(G_i)=\gamma$.

We prove that there exists a finite unit-distance graph in the plane with independence ratio strictly smaller than 1/4, answering a question of Erdős. Our proof closely follows the framework of Matolcsi, Ruzsa, Varga, and Zsámboki, based on the geometric fractional chromatic number, but adds a carefully chosen two-vertex augmentation that pushes their 27-vertex construction from geometric fractional chromatic number $4$ to a value strictly larger than 4. This disproves their Conjecture 1, and implies that the fractional chromatic number of the plane is strictly larger than 4. The proof can be made fully constructive, but the resulting finite graph has an enormous number of vertices.

The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $D(G)$ and $A(G)$ are the diagonal degree matrix and the adjacency matrix of $G$, respectively. The signless Laplacian spectral radius of $G$ is the largest eigenvalue of $Q(G)$. For a positive integer $t$, a graph is called $tK_3$-free if it contains no $t$ vertex-disjoint triangles. In this paper, for every fixed $t\geq 2$ and all $n\geq 28t-17$, we determine the unique graph achieving the maximum signless Laplacian spectral radius among all $tK_3$-free graphs of order $n$.

Let $\mathcal Y(z;t)$ be the modified Greaves–Jing–Zhu operator on the odd power-sum ring. We first point out that this operator can be obtained from the classical neutral operator by a simple diagonal change of variables. We then study products in which the two deformation parameters are not necessarily the same. For two parameters $t$ and $s$, we compute the scalar factor that appears in the mixed product. This factor has an explicit exponential form and, in a completed setting, can also be written as a quotient of infinite $t$-Pochhammer products. We also give a recurrence for its coefficients, a product formula for several mixed operators, and formulas for the coefficients obtained after applying the operators to $\mathbf 1$. A particularly simple case occurs when $s=t^M$. In this case the scalar factor becomes the finite quotient $(u;t)_M/(-u;t)_M$. Its coefficients are signed principal specializations of one-row Schur $Q$-functions. As a result, after removing the signs, these coefficients are nonnegative palindromic polynomials. We also give a Gaussian-binomial formula and a finite-order recurrence.

Let $q$ range over odd prime powers and let $G_q=\mathrm{GL}_2(\mathbb{F}_q)$. Fix a prime number $\ell$. Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\ell$, in the sense of divisibility in the ring of algebraic integers. We prove that $N_\ell(q)=\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$ for every $\epsilon>0$, where $N_\ell(q)$ denotes the number of entries which are not divisible by $\ell$. We also show that the number of zero entries is $\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$. Consequently, the proportion of all entries not divisible by $\ell$ tends to $1/2$, while the proportion of nonzero entries not divisible by $\ell$ tends to $1$. This differs significantly from the symmetric-group case, where almost every character-table entry is divisible by any fixed prime. We also prove an angular equidistribution result for the nonzero character values as $q\to\infty$. We show that the arguments become equidistributed in $[0,2\pi]$. This proves an analogue of Miller's question on the distribution of signs among the nonzero entries in character tables of symmetric groups.

The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a simplicial complex. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective version of the nerve theorem, yielding a new and powerful tool for homology computation. The essence of our theorem can be formulated in the following manner. Suppose, $X$ is a simplicial complex with covering subcomplexes $A_1, \dots ,A_k$, that is, $X= \cup_{i=1}^k A_i$ and $\mathcal{N}(X)$ is the nerve of $X$ with respect to its covering. Let $\mathcal{W}_\alpha$ be a given gradient vector field on $A_{\alpha}(=\cap_{i \in \alpha} A_i)$ for each $\alpha \in \mathcal{N}(X)$. Then, we use the mere information of the gradient trajectories in $A_{\alpha}$ for each $\alpha \in \mathcal{N}(X)$ to explicitly compute the homology groups of $X$. Furthermore, we point out here, that these gradient vector fields do not need to be coherent, that is, they do not need to coincide on the intersections, which gives us ample flexibility to apply our theorem. Moreover, we can further simplify the computation of the homology groups using a gradient vector field on the nerve of $X$. Our approach is purely combinatorial, in the sense that it does not involve any notions of geometric realisation, continuity or homotopy, which makes it more amenable to computation and coding.

Erdős conjectured that every triangle-free graph on $N$ vertices can be made bipartite by deleting at most $N^2/25$ edges; the bound would be sharp, attained by the balanced blow-up $C_5[N/5]$. Writing $\beta(G)$ for the minimum number of edges whose deletion makes $G$ bipartite and $a(N) = \max\{\beta(G):G$ triangle-free on $N$ vertices$\}$, the conjecture is $a(N)\le N^2/25$, and for $N=5n$ it reads $a(5n)\le n^2$. Balogh, Clemen and Lidícký proved it for large $N$ in the two density tails (edge density at most $0.2486$ or at least $0.3197$) and proved the global bound $a(N)\le N^2/23.5$; the medium-density band remains open. We prove \[ a(5n) = n^2 \qquad \text{for every } 1 \le n \le 40, \quad \text{i.e. } N \in \{5,10,\dots,200\}. \] The proof is computer-assisted and combines three ingredients. (i) A per-root-MaxCut envelope: for the $107$ triangle-free $7$-root types, the mean over types of the best per-type cut is an upper bound $d_{\rm mono}(W)\le U_7(W)$ that is tight at the $C_5$-blow-up. (ii) An order-$10$ flag-algebra certificate – the per-root-MaxCut rows at $7$ and $8$ roots together with rooted-Horn cuts and a manifestly-PSD moment block – bounds the envelope on the medium band, $U_7(W)\le \tfrac{2}{25}+\delta$ with an explicit rational $\delta\approx 4.8558\times10^{-5}$, for every triangle-free graphon $W$ of edge density in $[0.2486,0.3197]$. (iii) The blow-up identity $\beta(G[t])=t^2\beta(G)$ plus integrality of $\beta$ turns this into $\beta(G)\le n^2+\tfrac{25}{2}n^2\delta$ for any $5n$-vertex band-density $G$, and $\tfrac{25}{2}n^2\delta<1$ for $n\le 40$; the two density tails are handled by the Balogh-Clemen-Lidícký bounds, transferred to finite $N$ by the same blow-up. The envelope bound $d_{\rm mono} \le U_7$ is a genuine graphon upper bound (each per-root rule is one global $2$-colouring), the certificate is verified in exact rational arithmetic, the moment positivity is Razborov's flag-algebra theorem exhibited as an exact Gram factorization, and the bound is cross-checked against brute-force max-cut on all triangle-free graphs of order at most $12$. The same envelope at orders $9$ and $10$ provably does not reach the constant needed for larger $n$; we explain why, and locate the all-$n$ conjecture at a single self-tight obstruction.

Graphs, maps on surfaces, and abstract polytopes are related combinatorial structures that tend to be studied by different communities using their own tools and databases. Maniplexes provide a unifying framework that captures all of them. A single database built around maniplexes would help researchers recognize shared structures and translate results across fields. Here we present a compact, interoperable format for storing maniplexes as edge-labeled graphs, designed with such a database in mind. As a first step, we connect two existing datasets of regular 4-maniplexes to the House of Graphs and to Potočnik's tetravalent graph censuses, using canonical forms of their flag graphs, 1-skeleton graphs, and 1-coskeleton graphs.

Given \(H\leq G\) finite abelian groups, a transversal \(T\subseteq G\) for \(G/H\) has fixed size \(|G/H|\), but its ambient difference support \(D(T)=T-T\) can vary with the embedding of \(H\) in \(G\). We call $ \delta(G,H)=\min_T |D(T)| $ the transversal difference number of the pair \((G,H)\). This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption. We prove various results regarding this invariant, including a general lower bound $\delta(G,H)\geq 2|G/H|-m(G,H), $ where \(m(G,H)\) is the largest order of a subgroup of \(G\) disjoint from \(H\). The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors. These results isolate the first genuinely new residual obstruction, namely the same-prime square plane \[ G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG. \] For odd \(p\), this case is the technical core of the paper. Here transversals are graphs of functions \(\mathbb F_p^2\to \mathbb F_p^2\), and \(D(T)\) decomposes into carry-corrected finite-field derivative images. We conjecture that \[ \delta(G,H)=(2p-1)^2 \] for all odd primes \(p\), prove the unconditional lower bound \(3p^2-p-1\), and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.

We study the inhomogeneous random graph with preferential attachment kernel and degree distribution with power-law exponent $\tau\in(2,3)$ as a representative of the class of graphs of preferential attachment type with infinite variance degrees. Under bond percolation with a positive retention probability independent of the size $n$ of the graph there is a unique macroscopic component with high probability. We therefore investigate percolation probabilities $p_n\downarrow0$. We identify a moving critical window at $p_c \sim \beta n^{(\tau-3)/(2\tau-2)}$. Above this window, when $p_n \gg p_c$, the maximal component has size of order $n p^{_{(\tau-1)/(3-\tau)}}_{_n}$ and it is unique. Below this window, when $n^{1/(1-\tau)} \ll p_n \ll p_c$, it is non-unique, star-shaped and has size of order $n^{1/(\tau-1)} p_n$. In the critical window itself, the largest component scaled by $\sqrt{n}$ converges in distribution to a positive random variable with a law given in terms of a subcritical Norros-Reittu graph. This behaviour is markedly different from that seen for other classes of scale-free graphs and is conjectured to persist throughout the broad class of growing graphs with infinite variance.

This article investigates a remarkable combinatorial identity involving a distinguished family of matrices whose entries are defined via binomial coefficients. Specifically, we consider a class of \( n \times n \) matrices parameterized by a positive integer \( m \), where each entry reflects a structured pattern derived from Pascal's triangle, particularly the diagonals corresponding to figurate numbers such as triangular, tetrahedral, and higher-dimensional simplex numbers. We establish, by means of a bijective argument, that the determinant of any such matrix is identically equal to \( 1 \), independent of the specific values of \( m \) and \( n \), provided that \( 2 \leq m \leq n \). This result unveils a profound connection between classical binomial identities and the enumeration of lattice paths in grid graphs.

A quantum Latin square of order \(n\), denoted by \(\operatorname{QLS}(n)\), is an \(n \times n\) square whose entries are unit column vectors in the \(n\)-dimensional Hilbert space \(\mathcal{H}_n\), such that each row and each column forms an orthonormal basis of \(\mathcal{H}_n\). The cardinality of a QLS($n$) is the number of distinct vectors up to a global phase in the array. A \(\mathrm{QLS}(n)\) whose main diagonal and anti-diagonal each forms an orthonormal basis of \(\mathcal{H}_n\) is called a diagonal quantum Latin square (\(\mathrm{DQLS}(n)\)). In this paper, we focus on the existence of the \(\mathrm{DQLS}(n)\) with maximum cardinality ($\mathrm{MCDQLS}(n)$). By employing direct constructions based on row-quantum Latin rectangle and special complete mapping, together with the recursive techniques such as the singular direct product construction, We have almost completely determined the existence of \(\mathrm{MCDQLS}(n)\), except for a few exceptional cases. This result is based on the study of the existence of idempotent \(\mathrm{QLS}(n)\) with maximum cardinality (\(\mathrm{MCQLS}(n)\)), and implies an existence result for pandiagonal quantum Latin squares with maximum cardinality (\(\mathrm{MCPQLS}(n)\)).

Negami's Planar Cover Conjecture asserts that a connected graph has a finite planar cover if and only if it can be embedded on the projective plane. While this statement has already been proven for rotation compatible planar covers, namely covers equipped with a certain condition on the rotation system, the existing proof relies on advanced algebraic and topological methods. In this paper, we provide another proof of this result, focusing primarily on combinatorial arguments based on a structural analysis with respect to a spanning tree in the base graph.

Slater's list of Rogers-Ramanujan type identities remains a central source of striking series-product formulas in the theory of partitions and basic hypergeometric series. Although many of these identities admit elegant analytic proofs through Bailey pairs, Bailey chains, or transformations of basic hypergeometric series, the partition-theoretic meaning of their series sides is often much less apparent. In this paper, which continues the program initiated in arXiv:2603.14179, we apply Andrews' theory of separable integer partition classes to further identities from Slater's list. We construct a strict overpartition class and two families of overpartitions with positional gap conditions, in which overlining is permitted only at alternating positions. Their multivariate generating functions give natural refinements of the series sides of Slater's identities (12), (28), (29), (47), (48), (50), and (51). We then use Heine-type transformations, a limiting form of Heine's transformations, Watson's $q$-analogue of Whipple's theorem, and classical theta-product identities to obtain alternative series representations and recover the associated products. In addition, we derive a new companion identity to a Slater identity and a signed companion formula. Our results further demonstrate that SIP classes provide a flexible framework for converting basic hypergeometric series into structured partition generating functions, while simultaneously producing refinements, transformations, and new Rogers-Ramanujan type identities.

Enami and Maezawa give a complete characterization of $(s_1, s_2, \ldots, s_k)$-linked planar graphs for any $k$-tuple of positive integers. In this paper, we investigate linkage problems for optimal 1-planar graphs. In particular, we show that every optimal 1-planar graph with connectivity $6$ is $(5, 5)$-linked. Moreover, for an optimal $1$-planar graph $G$ that is not $(2,2,1)$-linked, we characterize disjoint vertex subsets $S_1, S_2, S_3$ in $G$ with $|S_1|=|S_2|=2$ and $|S_3|=1$ such that $G$ is not $\{S_1,S_2,S_3\}$-linked.

This work introduces and studies strong affine semigroups, extending the notion of strong numerical semigroups to the higher-dimensional setting. We show that non-numerical strong affine semigroups present structural differences with respect to strong numerical semigroups. Special attention is devoted to strong $\mathcal C$-semigroups. We prove that the family of strong $\mathcal C$-semigroups with a given set of multiplicities $E$ admits a maximal element and has a tree structure. We characterize when this family is finite and provide an algorithm to compute all such semigroups up to a fixed genus. We also introduce the notion of special strong affine semigroups and obtain refined versions of several previous results. Finally, we study toric ideals arising from strong affine semigroups, determining their indispensable monomials and Betti elements for several families.

Let $R(z)=\sum_{n=0}^{\infty} r_n z^n$ be a power series with $|r_n|=1$ for every $n\ge 0$. We show that for each integer $m\ge 2$, the coefficient sequence of $R(z)^m$ is unbounded. The proof combines Parseval's identity with Jensen's inequality. As a consequence, Conjecture~3.9 of Gawron, Miska, and Ulas \cite{gmu} is confirmed.

For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real Schur roots of the associated root system. We study the two-coordinate projections $(c_v, c_w)$ of this configuration: when the difference $c_v - c_w$ is bounded, the image lies in a finite band of lattice lines, and we ask when the projection fills every lattice point of that band. In affine type, boundedness is equivalent to $\delta_v=\delta_w$ for the null root $\delta$. Writing this common coordinate as $r$, we prove that a band line fills exactly when its transjective root classes cover every residue modulo $r$. This yields a complete classification for every acyclic orientation of every simply-laced affine diagram. For a source-sink $\widetilde A_n$ quiver, every projection fills except the source-sink diagonal. In tree type, every width-one band fills, while every width-two band fails on its two boundary lines because of a congruence gap. The Auslander–Reiten defect further determines whether such a boundary line is finite or contains an infinite arithmetic progression; finiteness is governed by a balanced-geodesic criterion. On the annulus, the absolute defect is the crossing number with the core curve, and the $\delta$-shift is the Dehn twist along it.

A Class-Uniformly Resolvable Design (CURD) is a resolvable design in which each parallel class has the same block structure. We study CURDS in which each parallel class contains one block of size $m$ and the remaining blocks have size $2$, for $m \ge 3$. In addition to establishing necessary conditions for such a CURD to exist, we present two general constructions. The first transforms a particular type of cyclic design with block size $k$ into a CURD with partition $m^12^{\frac{n-m}{2}}$ where $m = 2k$. This construction is used to generate CURDS with 26 varieties (where $m=6$) and with 82 varieties (where $m=10$). The second constructs a CURD with partition $m^12^{\frac{n-m}{2}}$ for every value of $m$ that is the power of an odd prime.

For $k\geq 1$, we prove that \[ [q^n z^s]J_k(z,q)\geq 0, \qquad (n\geq 0,\ s\in\mathbb Z) \] for the normalized Jacobi triple product tails \[ J_k(z,q) = \frac{ \sum_{j=k}^{\infty}(-1)^{j-k} q^{\binom{j+1}{2}}(z^{-j}+\cdots+z^j)} {(zq,q/z;q)_\infty}. \] This result not only implies Merca's stronger nonnegativity conjecture on truncated Jacobi triple product series in full generality, but also yields infinite families of linear inequalities for two-colored partitions and partitions with parts in the residue classes $\pm S \pmod{R}$. We present a combinatorial proof wherein a sign-reversing involution reduces the normalized Jacobi triple product tails to the invariant subsets according to the generalized minimal-excludant of partitions. Furthermore, by combing an invertible lift operator on Frobenius arms with Konan's size- and length-preserving bijection, an injection is constructed between the consecutive invariant subsets, which implies the coefficientwise positivity of the normalized Jacobi triple product tails.

The term tropical pseudonorm refers to a family of (not necessarily symmetric) gauge functions that arise in tropical or idempotent geometry. An important characteristic of these gauges is their invariance under translation by a constant vector, allowing them to descent naturally to tropical projective spaces. In this work, we explore the tropical one-infinity pseudonorm, a polyhedral hybrid gauge that allows for tunable asymmetry, in the context of a Fermat–Weber location problem. We extend previous formulations in considering non-finite data, and we investigate several variants of the inverse problem, providing linear programming formulations for their solution.

In a recent paper, Antić et al.~studied collections of planar orthogonal drawings of a graph where every edge is unbent in at least one drawing. This paper generalizes this concept to non-planar drawings, and shows that then two drawings always suffice (for planar drawings three drawings are sometimes needed). The results can also be generalized to $s$-grid drawings for $s\geq 3$.

We study intersections of conjugacy classes of square matrices over a finite field with affine coordinate subspaces, or equivalently matrices in a fixed adjoint orbit with prescribed entries. Our main result treats the case of prescribed columns: for a partially defined linear map we give a Hall scalar product formula for the number of extensions to an endomorphism with prescribed similarity invariants. This formula is expressed in terms of skew modified Hall–Littlewood functions and $q$-Whittaker functions. As applications, we count monic matrix polynomials over $\mathbb{F}_q$ with prescribed Smith normal form and with prescribed determinant, and recover the Gerstenhaber–Reiner formula for the number of square matrices with a fixed characteristic polynomial. We also note that known point-count formulas for Hessenberg varieties imply related formulas for Hessenberg supports involving chromatic quasisymmetric functions, motivating polynomiality questions for more general supports and prescribed affine slices.

In this paper, we study totally disjoint diametral paths in simple connected graphs. A diametral path in a graph is a shortest path that connects two vertices whose mutual distance is equal to the diameter of the graph. Totally disjoint paths are paths that have no vertices in common, including their end vertices. We show that the problem of deciding whether a graph $G$ has $k$ totally disjoint diametral paths is NP-complete. We consider restricted classes of graphs for which the problem of determining the maximum size of a set of totally disjoint diametral paths is readily solved. We then give a linear-time algorithm for a subclass of maximal outerplanar graphs called 2-paths, define a polynomial-time algorithm for threshold graphs, and establish a structural bound for proper interval graphs. Finally, we define classes of extremal graphs with $k$ totally disjoint diametral paths of length $d$ having the fewest possible number of edges.

We study the distance between two uniformly chosen points on a uniform random triangulation whose genus g is proportional to the number of faces 2n. We show that the distance rescaled by log(n) converges in probability to a deterministic constant, which answers a conjecture of Budzinski, Chapuy and Louf. The proof relies on the precise study of the volume growth of the ball of radius r for r of order log(n). The main ingredients are the recent local convergence results for uniform triangulations with boundaries and the isoperimetric inequalities obtained by Budzinski and Louf.

Given a finite nonempty planar point set $S$, what is the maximum number of congruent copies of $S$ contained in a set of $n$ points in the Euclidean plane? Building on OpenAI's recent breakthrough on the unit distance problem, we construct planar sets consisting of $n$ points that contain $\Omega_S(n^{1+\delta_S})$ congruent copies of $S$, for some positive constant $\delta_S$ depending only on $S$. This answers a question of Brass and Pach in a strong form, and makes progress on questions posed by Erdős and Purdy, and Ábrego and Fernández-Merchant. Our proof uses the number field construction from Sawin's quantitative refinement of OpenAI's result and consequently yields an explicit choice for $\delta_S$ for each fixed $S$.

For a graph $G$ and a field $\mathbb{F},$ the first clique-homology $H_1(\text{Cl}(G),\mathbb{F})$ vanishes precisely when the cycle space of $G$ over $\mathbb{F}$ is generated by the signed boundaries of the triangles. We develop cycle-surgery methods for establishing this property in arbitrary characteristic and apply them to strongly regular graphs. Combining our results with Neumaier's classification, we show that $H_1(\text{Cl}(G),\mathbb{F}) \ne 0$ can only occur in the Petersen graph, the Shrikhande graph, the complete bipartite graphs, the conference graphs on at most $255$ vertices, the lattice graphs, and the finite exceptional families $E_m$ in Neumaier's classification of strongly regular graphs with smallest adjacency eigenvalue $-m$, for some integer $m \geq 3$. Consequently, if $(G_n)_{n\geq 1}$ is an infinite family of pairwise distinct strongly regular graphs and $(\mathbb{F}_n)_{n\geq 1}$ is a sequence of fields such that $H_1(\text{Cl}(G_n), \mathbb{F}_n)\not=0$ for every $n$, then either $G_n$ is a lattice graph for infinitely many $n$, or $\lim_{n\rightarrow +\infty} \lambda_{\min}(G_n)=-\infty$. For Latin square graphs, we determine the clique homologies over arbitrary fields and show that if $G$ is the strongly regular graph associated with a Latin square $M$ of order $n \geq 5$ and $\mathbb{F}$ is any field, then $H_i(\text{Cl}(G),\mathbb{F})=0$ for $i=1$ or $i \geq 3$, and $\dim H_2(\text{Cl}(G),\mathbb{F})=(n-1)^3-I(M),$ where $I(M)$ is the number of $2 \times 2$ Latin subsquares or intercalates in $M$.

We construct a category of vector space-like objects called linear spaces over perfect idylls $k$ (an algebraic structure generalizing fields and hyperfields), which is a specialization of modules over $k$. Previous authors have noted that naive linear algebra in the product $k^n$ fails because linear independence does not satisfy matroid independence axioms. We give a sufficient axiom so that linear independence in $k$-linear spaces does satisfy matroid independence axioms. We also examine basic categorical properties of $k$-linear spaces. In particular, the category of $k$-linear spaces has no products, explaining the failure of naive linear algebra in $k^n$. Furthermore, we explore the categorical relationship between linear spaces over a perfect idyll, ordinary matroids, and matroids over a perfect idyll, connecting the existing theories of matroids and modules.

We study the $R$-sequenceability of finite groups of odd order. Building on the classical theory of $R^*$-sequences and orthomorphisms, we explore two tools: the notion of $R^{**}$-sequenceability, a strengthening of $R^*$-sequenceability tailored for inductive arguments over normal subgroups with cyclic quotients, and the odd cycle index $\tau(G)$, which measures how many orthomorphisms are required to generate a full cycle together with an involution. Our main result is a Quotient-Normal Gadget theorem, which shows that if $G$ has a normal subgroup $N$ such that $G/N$ is $R^{**}$-sequenceable and $\tau(N) \leq |G/N| - 3$, then $G$ itself is $R^{**}$-sequenceable. We prove that $\tau(G) = 2$ for cyclic groups of order coprime with $3$, and establish an inductive bound $\tau(G) \leq \max\{\tau(N), \tau(G/N)\}$ for odd ordered groups with a normal subgroup $N$. As consequences, we show that every group whose order is coprime with $30$ is $R$-sequenceable, and that every nilpotent group whose order is coprime with $6$ and not a power of $5$ is $R$-sequenceable. These results extend prior work on abelian groups to broad families of non-abelian groups.