Papers

For an edge-weighted graph $G=(V,E)$ and a stretch parameter $t\geq 1$, a $t$-spanner is a subgraph $H\subseteq G$ such that the shortest path distances in $G$ and $H$ satisfy $\delta_H(u,v)\leq t\, \delta_G(u,v)$ for all $u,v\in V$. In metric spanners, $V$ is a finite metric space, and $G$ is the complete graph with edge weights corresponding to the distances between the endpoints. When $G$ is the complete graph on $n$ points in the plane, $O(n)$-size $t$-spanners are possible for any $t>1$: For every $\varepsilon>0$, there is an $(1+\varepsilon)$-spanner with $O(n/\varepsilon)$ edges (i.e., the stretch can be arbitrarily close to 1). When $G=K(R,B)$ is the complete bipartite graph on $n$ bichromatic points in the plane, in general, no spanner construction can guarantee stretch $t<3$ with $o(n^2)$ edges. Bose et al.~(SICOMP 2009) constructed a $(3+\varepsilon)$-spanner with $O(n\log n)$ edges for any constant $\varepsilon>0$. Our main result is a new construction for a $(3+\varepsilon)$-spanner with $O(\sqrt{1/\varepsilon}\cdot n)$ edges. Eliminating the $O(\log n)$ factor resolves a problem left open for more than 17 years, and raises a new research problem about optimizing the dependence on $\varepsilon$. We also study spanners for $G=K(R,B)$ on $n$ bichromatic points on the real line: In this case, we show that the MST of $K(R,B)$ is a 7-spanner, and we construct a 3-spanner with at most $2n-3$ edges.

Fix a prime $p$ and a constant $\frac{1}{p}<\alpha\leq 1$. Consider the random Erdős–Rényi bipartite graph $G_{\alpha}(n,u)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<u<1$. The authors of [1] and [8] conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $G_{\alpha}(n,u)$ as $n\to\infty$. We prove this conjecture for odd primes $p$. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group to $H$, for each finite abelian $p$-group $H$. However, in our setting, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this issue by discarding the graphs for which too many vertices have degrees divisible by $p$. Once we remove the contribution of this rare set of graphs, then the surjective moments converge to the expected values. When $p$ is odd, applying Wood's universality theorem yields the desired convergence in distribution. For $p=2$, our computed moments (after excluding the rare set of graphs) match those of the conjectured distribution. However, these moments do not uniquely determine a distribution.

Green's popular difference theorem says that for every \(\varepsilon>0\), all sufficiently large primes \(p\), and every set \(A\subseteq\mathbb F_p\) of density \(\alpha\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq \alpha^3-\varepsilon. \] We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every \(\varepsilon>0\), all sufficiently large primes \(p\), and every set \(A\subseteq\mathbb F_p\) of density \(\alpha\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{\omega_i} \geq \alpha^{1+\sum_i\omega_i}-\varepsilon \] simultaneously for every \(\omega=(\omega_1,\dots,\omega_k)\in\{0,1\}^k\). We also show that such simultaneous popular difference phenomena have sharp limitations by proving that for every sufficiently large prime \(p\), there is a constant \(c>0\) such that, for all sufficiently large \(n\), one can find a set \(A\subseteq\mathbb F_p^n\) of density \(1/2+o_n(1)\) satisfying \[ \max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c. \] That is, the strengthening of Green's result, in this case over $\mathbb F_p^n$ for $p$ fixed and $n$ tending to infinity, requiring that both \(d\) and \(2d\) are simultaneously popular differences for three-term arithmetic progressions is false.

In this note we determine the maximum number of arcs in a digraph on $n$ vertices that does not contain a length-$k$ directed path $\overrightarrow{P}_{k+1}$ for $n \geq 50k^6$. This improves a theorem of Zhou and Li [$\textit{Graphs Combin.}$ 39(3), 2023] who proved the result with a superexponential threshold on $n$.

The paper studies the problem of circular total positivity of the symmetric matrices with zero row sums. These matrices are exactly response matrices of the electrical networks. Alman, Lian and Tran described tests for circular total positivity in two related frameworks: the cluster algebra $\mathcal{CM}_n$ and the Laurent Phenomenon algebra $\mathcal{LM}_n$. Our first result is the construction of a seed in Scott's cluster algebra structure on the coordinate ring of the Grassmannian $\mathrm{Gr}(n-1,2n)$ that consists entirely of circular minors. We compare the cluster structure induced by this seed with $\mathcal{CM}_n$. In particular, for odd $n$ the cluster algebra structure $\mathcal{CM}_n$ is isomorphic to the cluster algebra structure on $\mathrm{Gr}(n-1,2n)$ subject to natural freezing and trivialization of certain cluster variables in their initial seeds. We use this isomorphism to relate circular total positivity to positivity in the Grassmannian. Our second result is that the Laurent Phenomenon algebra $\mathcal{LM}_n$ is isomorphic to the coordinate ring of the noncompactified space of electrical network, or equivalently, to a certain localization of the grove algebra.

Extremal matroids are those that yield vertices in the polytope of matroids. We show how to get the point, in the appropriate polytope of matroids, for the free product $M\square M'$ of matroids $M$ and $M'$ from the points for $M$ and $M'$ in their polytopes of matroids. With this we show that if $M\square M'$ is extremal, then $M$ and $M'$ are extremal. The converse is false. We identify a large class of matroids that includes perfect matroid designs and sparse paving matroids, and we find sufficient conditions under which free products of extremal matroids in this class are extremal.

The permanent of an $n \times n$ matrix is the coefficient of the fully mixed monomial $x_0 \cdots x_{n-1}$ in the product of its row forms $\prod_i\bigl(\sum_j b_{ij} x_j\bigr)$, and the classical exact algorithms of Ryser and Glynn compute it by summing $2^{n-1}$ evaluations of that product over the Boolean cube. We recast this as a single principle – an evaluation scheme together with an orthogonality criterion that decides, in one line, whether its weighted sum equals the permanent – and show that Ryser's formula, Glynn's formula, and a discrete Fourier transform are three instances of it: the $0$–$1$ and $\pm 1$ Hadamard systems, and a single cyclic transform of length $N$. The DFT scheme evaluates $\mathrm{per}\,B$ as one coefficient of a univariate polynomial modulo $x^N - 1$; it is exact exactly when the exponent set is valid mod $N$, runs in the same $\Theta(2^n n)$ operations as the classical formulas, and over a finite field is a number-theoretic transform that returns the exact integer permanent by the Chinese remainder theorem. Where the Hadamard schemes are orthogonal for free, the cyclic scheme trades this for an arithmetic condition on the exponents; the smallest length at which it can be met is $N = 2^n - 2^{\lfloor \log_2 n \rfloor}$, established in the companion paper arXiv:2607.08366. All three are character sums over a finite abelian group, and the criterion holds precisely for such groups.

We construct an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality $8$ with full affine span, together with an $F$-tiling $T$, such that the orbit closure of $T$ in $\{0,1\}^{\mathbb{Z}^2}$ does not contain a $1$-periodic $F$-tiling. Since every $F$-tiling is a low-complexity configuration with respect to the window $\bar F := \{-a : a \in F\}$, this supplies a "non-degenerate" counterexample, in a strong sense, to Nivat's conjecture for non-convex windows. This answers, in the negative, a question of Kari and Moutot (2023) whether every such counterexample must be degenerate, in the sense that the probing window is contained in a coset of a proper finite-index sublattice. We complement this with a positive result: for every exact cluster $F$ of full affine span whose cardinality is the square of a prime, every $F$-tiling has a $1$-periodic $F$-tiling in its orbit closure. Together with Szegedy's theorem that every tiling by a cluster of prime cardinality is $1$-periodic, this shows that no cluster of fewer than $8$ cells can exhibit the phenomenon, with the possible exception of cardinality $6$, which we leave open.

Let $G = (V,E)$ be a finite directed graph with a non-negative real length $\mu_e$ assigned to every directed edge $e \in E$. We assume that $\mu_e = +\infty$ for every non-edge $e \not\in E$. Fix any two distinct vertices $a, b \in V$. A directed path from $a$ to $b$ is called an $(a,b)$-path. An edge $e$ is called an $(a,b)$-bridge if it belongs to all $(a,b)$-paths. Furthermore, it is not difficult to show that all $(a,b)$-paths pass all $(a,b)$-bridges in the same order. Define the distance $\mu(a,b)$ from $a$ to $b$ as the sum of lengths of all $(a,b)$-bridges. Furthermore, $\mu(a,b) = \infty$ if there are no $(a,b)$-paths and $\mu(a,b) = 0$ if $(a,b)$-paths exist but there are no $(a,b)$-bridges. It is easily seen that $\mu(a,b)$ can be computed in polynomial time and the metric inequality $\mu(a,b) \leq \mu(a,c) + \mu(c,b)$ holds for every $a,b,c \in V$. Furthermore, equality holds if and only if each $(a,b)$-bridge is either an $(a,c)$- or a $(c,b)$-bridge. \newline We will show that this is a special limit case $r=s \rightarrow 0$ of the inequality $\mu(a,b)^{s/r} \leq \mu(a,c)^{s/r} + \mu(c,b)^{s/r}$ obtained for all positive real parameters $r$ and $s$ in the paper “Metric and ultrametric inequalities for directed graphs”, Discrete Appl. Math. 314 (2022) 93–104, along with 3 other limit cases $r=s \rightarrow \infty$, $r=1, s \rightarrow \infty$, and $s = 1, r \rightarrow 0$, considered in that paper.

The Erdős–Pósa theorem asserts that every graph $G$ with no $k$ disjoint cycles contains a set $X$ of $f(k)$ vertices such that $G\setminus X$ has no cycle. Robertson and Seymour showed that this Erdős–Pósa property also holds for $H$-minor models of any planar graph $H$. Equivalently, if $G$ has no $k$ minor models of $H$ pairwise at distance at least 1 (i.e. disjoint), then one can remove $f(k,H)$ balls of radius 0 (i.e. vertices) to make the graph $H$-minor free. We show that this coarse graph theory point of view generalizes to distance at least 2 versus radius 1 balls, yielding the induced Erdős–Pósa property for planar minors. Namely, every graph $G$ which does not contain $k$ pairwise non-adjacent minor models of a planar graph $H$ (we say that $G$ is $kH$-free) can be made $H$-minor free by removing $f(k,H)$ neighborhoods. The proof relies on the fact that sparse $kH$-free graphs have linearly many independent large protrusions. The same method gives that sparse $kH$-free graphs can be made $H$-minor free by deleting $O(\log n)$ vertices (and thus have logarithmic tree-width). This gives a quasi-polynomial algorithm for the Maximum Independent Set problem for $kH$-free graphs.

We study the computational complexity of problems that ask if a given graph admits an edge-coloring that does not contain an edge-colored clique from some fixed finite family. We show that every such problem is poly-time equivalent to a Constraint Satisfaction Problem, yielding a P vs. NP-complete dichotomy. Our main contribution lies in the reduction from the CSP to the coloring problem where we apply methods from Ramsey theory and a novel notion of cut-homotopy.

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized planar Turán number $\mathrm{ex}_\mathcal{P}(n, H, \mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex planar graph that contains no graph $F \in \mathcal{F}$ as a subgraph. When only induced copies of $H$ are counted, we denote the corresponding generalized planar Turán number by $\mathrm{ex}_\mathcal{P}(n, H^{\mathrm{ind}}, \mathcal{F})$. Győri and Karim determined $\mathrm{ex}_\mathcal{P}(n, C_{5}, \{C_3\})$. In this paper, we determine the exact value of $\mathrm{ex}_\mathcal{P}(n, C_{2k+1}, \{C_3,C_5,\ldots,C_{2k-1}\})$ for every $k \ge 3$. Since all shorter odd cycles are forbidden, every $C_{2k+1}$ is induced. This problem is closely related to the inducibility of odd cycles in planar graphs. Ghosh, Győri, Janzer, Paulos, Salia and Zamora~(and independently Savery) determined the exact value of $\mathrm{ex}_\mathcal{P}(n, C_5^{\mathrm{ind}}, \emptyset)$. Moreover, they established a conjecture for all odd cycles $C_{2k+1}$ with $k \ge 3$. Our result confirms their conjecture under the additional assumption that all shorter odd cycles are forbidden.

An open question asks whether the chromatic symmetric function (CSF) of a graph distinguishes non-isomorphic trees. While it is known that the CSF does not distinguish unicyclic graphs, examples of pairs of unicyclic graphs with the same CSF and girth larger than 3 were not known until very recently. This manuscript exhibits a sequence of pairs of non-isomorphic, connected, unicyclic graphs with increasing girth and which share the same CSF. This also provides the first infinite family of bipartite graphs with the same CSF. Our main technique is to apply a version of the "triple deletion" modular relation, due independently to Guay-Paquet and Orellana–Scott.

We construct a diamond-free family in the Boolean lattice whose size is asymptotically larger than the union of two middle layers. Denote the diamond poset by $Q_2$ and let $La(n,Q_2)$ be the maximum size of a family in $2^{[n]}$ containing no weak copy of $Q_2$. We prove $La(n,Q_2) \ge (c+o(1))\binom{n}{\lfloor n/2\rfloor}$, where $c \approx 2.147908$. In particular, this disproves the diamond conjecture.

We prove a divisibility theorem for the signed $J$-characteristics of two-level designs: if the number of factors $n$ is odd and every $J$-characteristic of a proper odd-cardinality subset of factors vanishes, then the top $J$-characteristic is divisible by $2^{n-1}$. As an arithmetic consequence, any two-level design whose $J$-characteristics vanish in orders one, two, three, five, and seven but which has a nonzero odd-order $J$-characteristic must have at least $256$ runs. This settles, uniformly in the number of factors, a conjecture of Eendebak, Schoen, Vazquez, and Goos (2023) on the nonexistence of certain strength-three even–odd designs with $56$ or $64$ runs. The divisibility bound is sharp at every odd order and is attained by the even-weight half-fraction.

A famous conjecture of Anton Kotzig states that for every even integer $n\ge 4$, the complete graph $K_n$ of order $n$ can be decomposed into $n - 1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. Despite the great interest, the conjecture is far from being solved. Here we show that the conjecture holds asymptotically, namely that $K_n$ can be decomposed into $n-1$ perfect matchings such that $(1-o(1))n$ of them have the property that any pair forms a Hamilton cycle.

We study the problem of realizing line configurations as collections of 2-spheres smoothly embedded in the complex projective plane. Building upon prior work by Ruberman and Starkston on topological realizations, we establish a stronger obstruction in the smooth category. Our proof relies on lattice-theoretic arguments based on Donaldson's diagonalization theorem.

We investigate constraints on the existence of topological and smooth realisations of combinatorial line arrangements and $(n_k)$-configurations in the complex projective plane. By replacing complex lines with locally-flatly or smoothly embedded 2-spheres, we explore the extent to which classical geometric results, such as Hirzebruch's inequality, persist in the topological or smooth category. We introduce two classes of special line arrangements that we call odd and even. We provide constraints for any smoothly realised, non-trivial, odd arrangement via Furuta's 10/8-Theorem. By looking at branched double covers and using the G-signature theorem, we study topologically realised, non-trivial, even arrangement. Finally, we establish a new lower bound for $(n_k)$-configurations, showing that for any topologically realised configuration we have $n \geq k^2-5$, which implies the non-existence of topological realisations for finite projective planes.

Let $G = (V,E)$ be a graph of order $n$. A bijection $f : V \rightarrow \{1,2,\cdots,n\}$ is a distance magic labeling of $G$ if there exists a positive integer $k$ such that $\sum_{u \in N(v)}f(u) = k$ for all $v \in V$, where $N(v)$ is the neighborhood of $v$. Any graph which admits a distance magic labeling is called a distance magic graph. In this article, we give a partial solution to the problem by Rao et al.[10] to predict all distance magic labelings of cartesian product of two cycles, $C_m \Box C_m$, where $m\equiv 2 \mod 4$. Further, we prove that the number of distance magic labelings of a distance magic graph is a multiple $| Aut(G)|$ where $Aut(G)$ is the automorphism group of the distance magic graph $G$.

Let $G$ be a simple graph of maximum degree $d$, and let $\mu(G)$ denote the largest eigenvalue of its Laplacian matrix. For a fixed integer $k\geq 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of $K_{1,k}$ satisfies $\mu(G)\leq (2 - \frac{2}{k} + o(1)) d$. We answer this question by proving the stronger sharp bound \[ \mu(G)\leq \left(2-\frac{2}{k}\right)(d+1). \] The proof combines a sign decomposition of a Laplacian Rayleigh vector with a weighted local Caro-Wei type inequality for independent sets.

We provide an elementary proof of a symmetric property of the top dimensional components of the quiver subvariety $A_n\dots A_1=0$.

Let $s_\mathbf{w}$ be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights $\mathbf{w}=(w_j)_{j\geq 0}$. We investigate Hankel determinants $\mathcal{H}_\mathbf{w}(n) = \det [s_{\mathbf{w}}(i+j)]_{0\leq i,j<n}$ and derive a general recursion that allows us to effectively compute $\mathcal{H}_\mathbf{w}(n)$ for all $n$. Applying it to the ordinary binary sum-of-digits, that is, $w_j=1$, we express $\mathcal{H}_\mathbf{w}(n)$ in a closed form for several sequences of indices, including the remarkably simple $$ \mathcal{H}_\mathbf{w}(\lceil 2^{k+2}/3\rceil)= (-1)^{\frac{(k+2)(k+3)}{2}}(k+1). $$ This yields an infinite family of explicit evaluations, giving a partial solution to a problem posed by Allouche and Shallit. Moreover, we closely study the specialization $w_j=t^j$, where the determinants become polynomials in $t$, and investigate their vanishing. For $t=2\zeta$, where $\zeta$ is a root of unity, we show that the determinants vanish on a large structured set of indices, while the complementary is sparse but infinite. In addition to $\mathcal{H}_\mathbf{w}(n)$, we consider Hankel determinants associated with the first difference of $s_{\mathbf{w}}$, obtaining an explicit product formula. This generalizes the results by Fokkink, Kraaikamp, and Shallit concerning Hankel determinants for the period-doubling sequence.

Let $p_1,...,p_k$ be a rationally independent sequence of integer valued nonlinear polynomials. We show that for all $E\subseteq \mathbb{N}$, every Folner sequence $\Phi$, and every $\varepsilon>0$, the set $$\left\{n\in \mathbb{N} : d_{\Phi}\left(E\cap(E+p_1(n))\cap\cdots\cap (E+p_k(n))\right) > d_{\Phi}(E)^{k+1}-\varepsilon\right\}$$ intersects every IP generated by a sequence with rational spectrum. Our methods involve the study of the characteristic factors for multiple ergodic polynomial averages along IPs. In particular, we also prove a pointwise convergence theorem for polynomial averages along IPs with rational spectrum, generalizing a well known result of Leibman.

The Irrelevant Vertex Technique is one of the cornerstones of algorithmic graph theory, underlying Robertson and Seymour's algorithm for Disjoint Paths and much of the algorithmic Graph Minors theory. We show that, in the setting of spanning routing, this technique exhibits an exact combinatorial limitation. Unlike classical routing problems, spanning routing is not governed by the number of distinguished vertices but by the way they are distributed throughout the graph. The input is a triple $(G,R,\mathcal{T})$ where $(G,R)$ is an annotated graph and $\mathcal{T}$ is a set of terminal pairs. The goal is to determine if $G$ contains a family of internally disjoint paths connecting the pairs in $\mathcal{T}$ such that the union of the paths spans the set $R$. We identify a new structural parameter of annotated graphs, called $\mathsf{depth}_2$, that measures precisely this phenomenon. Our main result is a complete combinatorial dichotomy: for every red-minor-closed class of annotated graphs, the Irrelevant Vertex Technique applies to Spanning Disjoint Paths if and only if the class has bounded $\mathsf{depth}_2$. Thus $\mathsf{depth}_2$ forms the exact structural boundary between classes where the Robertson-Seymour paradigm survives and those where it breaks down. Our proof combines a new local structure theorem for annotated graphs of bounded $\mathsf{depth}_2$ with a spanning analogue of the celebrated Vital Linkage Theorem. The resulting algorithm solves Spanning Disjoint Paths in time $2^{2^{\mathbf{poly}(k+d)}}\cdot n^2$ where $d$ is the $\mathsf{depth}_2$ of the input instance. We provide matching lower bounds showing that beyond bounded $\mathsf{depth}_2$ no irrelevant-vertex rule can exist, even on planar graphs. In particular, $\mathsf{depth}_2$ is the exact combinatorial barrier for the Irrelevant Vertex Technique under spanning constraints.

This paper introduces Bernoulli-type and Euler-type Mersenne-Lerch interpolation families associated with Apostol-Mersenne polynomial families. Their construction is based on the Mersenne translation polynomials \(P_{n,M}(x;m)\), defined by \[ e_M^{xt}(e_M^t)^m = \sum_{n=0}^{\infty}P_{n,M}(x;m)\frac{t^n}{M_n!}. \] Explicit formulas for these polynomials are derived, including an expansion in terms of M-Stirling numbers of the second kind. At nonpositive integers, the resulting interpolation values recover the Apostol-Mersenne-Bernoulli and Apostol-Mersenne-Euler polynomials of order \(r\). Derivative, integral, addition, and difference formulas are also obtained for these values. A comparison with \(q\)-calculus shows that the M-factorial, M-binomial coefficient, M-derivative, and M-exponential are obtained by setting \(q=2\) in the corresponding \(q\)-calculus expressions.

We present a polynomial-time algorithm for optimally coloring perfect graphs that is based entirely on graph-theoretic operations. At its core, the algorithm decides whether a perfect graph contains a clique of a given size by iteratively counting walks in the graph with certain weights assigned to its edges and nonedges. These weights are initialized according to a uniform scheme and then updated in each iteration based on the walk counts from the previous iteration.

The dimension of a poset $P$ is the minimum positive integer $d$ such that $P$ is an induced subposet of $\mathbb{R}^d$ equipped with the product order. We give a constant-factor polynomial-time approximation algorithm for computing dimension in the class of posets with a planar (Hasse) diagram. While computing the dimension of a poset is NP-hard in general, the computational complexity of the problem for planar posets remains open. The algorithmic result is driven by a structural understanding of the canonical obstruction to small dimension: standard examples. A longstanding problem, originating in the early 1980s, asked whether every poset with a planar diagram has dimension bounded by a function of the maximum order of a standard example that it contains. In the first paper of the series, we have resolved the problem in a more general setting of posets with planar cover graphs by establishing a polynomial bound. We prove a stronger bound in the original setting, namely, for every poset $P$ with a planar diagram $\mathrm{dim}(P) \leq 96\mathrm{se}(P)+672$, where $\mathrm{dim}(P)$ denotes the dimension of $P$ and $\mathrm{se}(P)$ denotes the maximum order of a standard example contained in $P$.

A vertex-transitive closure of $\Gamma$ is a vertex-transitive supergraph of $\Gamma$ on the same vertex set. The vertex-transitive number of a graph $\Gamma$, denoted by $d_{vt}(\Gamma)$, is the smallest integer for which there exists a $d_{vt}(\Gamma)$-regular vertex-transitive closure of $\Gamma$. In this paper we use various algebraic and combinatorial methods to study vertex-transitive closures of graphs and the associated vertex-transitive number.

A $\{P_2,P_3\}$-factor in a graph $G$ is a factor of $G$ in which every component is a path on two or three vertices. Let $T\Box P_n$ be the Cartesian product of a tree $T$ and a path on $n$ vertices. Kao and Weng proved that $T\Box P_n$ is hamiltonian if $T$ has a path factor and $n$ is a sufficiently large even integer. In this article we prove that, for every odd $n$, there exists a tree $T$ of maximum degree 4 that has a $\{P_2,P_3\}$-factor such that $T\Box P_n$ is not hamiltonian, thereby refuting a conjecture by Kao and Weng.

We study deformations of polytopes that preserve edge lengths and face coplanarities. This gives rise to a notion of rigidity, for which we develop a second-order theory together with an effective algorithm for testing second-order rigidity symbolically. The strength of these new tools is demonstrated on several polytope classes, including the previously intractable regular dodecahedron, which we find to be rigid. We also find that a single tested polytope evades our techniques and will therefore provide a simple test case for future developments of tools of even higher order. This paper also contains a study of so-called edge-length perturbations. We point out connections between rigidity and the ability to realize polytopes with slightly perturbed edge lengths. To explore these connections in practice, we dedicate a section to another case study of the regular dodecahedron. We investigate the local behavior of its realization space with a view towards edge-length perturbations, singularities and generic global rigidity.