Papers

The edge-girth of an edge $e$ in a simple connected graph is the length of a shortest cycle containing $e$, with $g_e = \infty$ if no such cycle exists, and the edge-girth sequence of a graph is the nondecreasing sequence of the edge-girths of its edges. We characterize the sequences that are realizable as the edge-girth sequence of a simple connected graph. A constant sequence $(g^{(m)})$ with $g$ finite and $m \geq 1$ is realizable if and only if $m = g$ or $m \geq \lceil 3g/2 \rceil$. Otherwise, writing $S = S_0 \uplus (g^{(m)})$, where $g$ is the maximum value of $S$ and $m$ its multiplicity, we prove that $S$ is realizable if and only if $S_0$ is realizable and $m$ lies in an explicit set determined by $g$ and by the maximum diameters of the graphs realizing $S_0$, alone or extended by fewer than $g$ edges of edge-girth $g$. The proof rests on a chaining lemma, which turns a shortest cycle of a realization into a lower bound on the maximum diameter of the realizations of a shorter sequence.

For integers $n\ge s\ge2$, let $e(n,s)$ be the maximum size of a family $\mathcal F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erdős–Kleitman problem, is closely related to the well-known Erdős matching problem. Frankl and Kupavskii posed a meta-conjecture predicting that the maximum is always attained by a weighted family. Fix $m\ge3$, write $n=ms+c$ with $0\le c<s$, and set $\ell=s-c$. For $0\le k\le m$, let $a_k=ms-kc-1$. For $A\in\binom{[n]}{a_k}$, define \[ \mathcal H^k(m,s,\ell;A):= \{F\subseteq[n]: k|F|+|F\cap A|\ge m(k+1)\}. \] This defines a unified class of weighted families with matching number less than $s$. Among these families, $\mathcal H^0$, $\mathcal H^1$, and $\mathcal H^m$ were previously known to be extremal in different ranges of $c$. We show that for $1\le k\le m-1$, all families $\mathcal H^k$ are uniquely extremal in some ranges of $c$. More precisely, we prove that for every $m\ge3$ and every $1\le k\le m-1$, there exist constants $\alpha=\alpha(m,k)>0$, $\beta=\beta(m,k)>0$ and an integer $s_0=s_0(m,k)$ such that, for all integers $s\ge s_0$ and all integers $c$ with $0\le c<s$, the only extremal families for $e(n,s)$ are the families $\mathcal H^k(m,s,\ell;A)$ with $A\in\binom{[n]}{a_k}$ whenever $\beta s^{(k-1)/k}\le c\le \alpha s^{k/(k+1)}$. In particular, this result determines an infinite number of new extremal families for the Erdős–Kleitman problem and verifies the Frankl–Kupavskii meta-conjecture in these ranges. This also provides a quantitative extension of the result of Kupavskii and Sokolov on the extremality of $\mathcal H^1$.

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, and let $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$. The projective Ore-degree of $\mathcal F$ is the minimum, over all $k$-subspaces $S\notin\mathcal F$, of the sum of the $\mathcal F$-degrees of the projective points contained in $S$. We prove sharp projective Ore analogues of the vector-space Erdős–Ko–Rado and Hilton–Milner theorems. The Ore–Erdős–Ko–Rado theorem holds for $n\ge2k+1$, with equality only for a full point-star. For nontrivial intersecting families, we determine the sharp Ore–Hilton–Milner threshold, together with the complete equality classification, when $q\ge3$ and $n\ge2k+1$, or when $q\ge2$ and $n\ge2k+2$. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s\ge3$ and $n\ge(2s-1)k-s+4$, and derive a multicolour Ramsey consequence.

The hypernatural numbers ${}^*\mathbb{N}$ of nonstandard analysis have recently proven to be an effective tool in arithmetic Ramsey theory. After introducing the fundamental “nonstandard" notions, we present several examples to illustrate the use of this technique in practice. In particular, we provide brief nonstandard proofs of some recent results concerning the monochromaticity of certain families of finite and infinite configurations.

We prove that the path $P_{11}$ does not admit uniform average mixing under any probability distribution on $\mathbb R$, answering a question of Baptista, Coutinho, and Marques in the negative. The proof is exact: we construct an explicit rational symmetric matrix $Y$ such that $\langle Y,M(t)\rangle_F=1$ for every $t\in\mathbb R$, whereas $\langle Y,J/11\rangle_F=12/11$.

Let $d \geq 1$ be an integer, $G$ be an abelian group and $\nu, w_1, \dots, w_{2d}: G \to [0, \infty)$ be functions with finite, non-empty supports. Define the generalised additive energy \[ E_{2d, \nu}(w_1, \dots, w_{2d}) = \sum_{y,y' \in G}\sum_{a_1, \dots, a_{2d} \in G } w_1(a_1) \dots w_{2d}(a_{2d}) \nu(y) \nu(y') 1_{\sum_{i=1}^d (a_i - a_{i+d}) = y-y'} .\] Moreover, for every $1 \leq i \leq 2d$, let $E_{2d, \nu}(w_i) = E_{2d, \nu}(w_i, \dots, w_i)$. A standard Fourier analytic argument delivers the estimate \[ E_{2d,\nu}(w_1, \dots, w_{2d}) \leq \prod_{1 \leq i \leq 2d} E_{2d, \nu}(w_i)^{1/2d}.\] In this note, we present a purely combinatorial proof of the above inequality. In particular, our proof does not use any Fourier or spectral analysis and relies on repeated applications of Cauchy–Schwarz inequality combined with a discrete convexity extension type argument. We also record a variation of this upper bound in the non-abelian setting via spectral inequalities following work of Hatami on graph norms, as well as a relevant sumset analogue obtained via iterative applications of the Plünnecke–Ruzsa inequality.

We prove that the local limit of the minimum spanning arborescence in the complete graph (which is an oriented cousin of the minimum spanning tree in the complete graph) is the same as that of the uniform random tree, oriented towards infinity. The latter is known to be the critical Poisson Galton–Watson tree conditioned to survive. This is in sharp contrast with the local limit of the minimum spanning tree, which is known to be different from that of the uniform spanning tree (due to results of Addario–Berry \cite{AddarioBerry2013} and Addario–Berry, Griffiths and Kang \cite{PWIT_local_minimum}). Thus we demonstrate that introducing orientations changes the local geometry of the minimum spanning tree in a non-trivial manner.

Flow polytopes of directed acyclic graphs form a central class of lattice polytopes in algebraic, geometric, and enumerative combinatorics. Order polytopes are one of the best understood families of lattice polytopes; their Ehrhart theory, triangulations, volumes, and face structures are closely controlled by the combinatorics of the underlying posets. Mészáros–Morales–Striker proved that the flow polytope of an $st$-planar directed acyclic graph is unimodularly equivalent to an order polytope. In this paper, we prove a converse after contracting idle edges. More precisely, for a directed acyclic graph $G$ with a unique source and a unique sink, let $\widetilde G$ be the graph obtained from $G$ by successively contracting idle edges until none remain. We prove that $\mathcal{F}(G)$ is unimodularly equivalent to an order polytope if and only if $\widetilde G$ is $st$-planar. In addition, under a local three-good-neighbor condition, we prove that for a directed acyclic graph with a unique source, a unique sink, and no idle edges, the graph is $st$-planar if and only if it avoids an explicit list of forbidden butterfly minors.

We disprove Hoàng's conjecture that a minimally nonperfectly divisible graph cannot contain a bisimplicial vertex by constructing an explicit infinite family. For every integer $t\geq1$, the graph $G_t$ in this family has clique number three, contains a bisimplicial vertex of degree four, and satisfies \[ |V(G_t)|=93+30(t-1),\qquad |E(G_t)|=320+104(t-1). \] In particular, the members are pairwise nonisomorphic. The construction uses a fixed 15-vertex rooted graph and a variable auxiliary graph. Every copy of the rooted graph forces its identified root into the perfect part of every perfect division. Three induced odd holes in the auxiliary graph then force a triangle into the other part. A uniform assignment lemma and a rooted product lemma show symbolically that every proper induced subgraph of every $G_t$ is perfectly divisible. The finite properties of the fixed rooted graph are verified by exact exhaustive computation, with an independent implementation providing a cross-check. The construction also gives an infinite family of negative examples to a prescribed-vertex problem of Hu, Xu and Zhuang.

The orientable genus polynomial of a graph counts its cellular embeddings by genus. For finite simple $2$-connected cubic graphs it is a cycle-matroid invariant: $M(G)\cong M(H)$ implies $\Gamma_G=\Gamma_H$. The adjacency spectrum and the genus polynomial are incomparable: neither determines the other. We exhibit connected cubic graphs on $16$ vertices sharing the adjacency spectrum, spanning-tree count, girth, diameter, vertex and edge connectivity, automorphism-group order, and cycle counts through length $10$, yet with pairwise distinct genus polynomials. Splitting the expected face count at twice the girth explains the difference: short faces are spectral, long faces are not. We construct an explicit infinite family of connected cospectral cubic pairs $(G_t,H_t)$ on $14+2t$ vertices whose minimum genera differ. We also compute the genus polynomials of all $7,875,918$ connected cubic graphs through $22$ vertices and derive from short-cycle counts a deterministic lower bound on the minimum genus.

A hole is an induced cycle of length at least four, and an even hole is a hole of even length. A cap is obtained from a hole by adding a vertex adjacent to exactly two consecutive vertices of the hole. Chen, Xu, and Xu proved that every $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graph $G$ satisfies $\chi(G)\leq \left\lceil\frac{5}{4}\omega(G)\right\rceil$, and improved this bound to $\chi(G)\leq \left\lceil\frac{7}{6}\omega(G)\right\rceil$ when $5$-holes are also excluded. They asked whether, for every integer $q\geq3$, every $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graph $G$ with no odd hole of length at most $2q-1$ satisfies $$ \chi(G)\leq \left\lceil\frac{2q+1}{2q}\omega(G)\right\rceil. $$ We answer this question affirmatively and show that the bound is sharp for every $q\geq3$.

For integers $k\le n$, let $L_{k,n}$ be the set of words over $[n]$ of length at most $k$ in which no symbol is repeated. We present a nondeterministic finite automaton (NFA) of size $3.918^k n^{O(1)}$, improving on the $4^{k+o(k)}n^{O(1)}$ construction of Ben-Basat, Gabizon, and Zehavi. Our proof organizes several classical ingredients—product automata, hashing, and coefficient estimates—into a gadget-amplification framework: We take the product of many copies of a small local NFA gadget, whose language is a subset of $L_{r,c}$, and hash the $k$ input symbols to copies and local colors. The hash family guarantees that, for every repetition-free input, some hash sends at most $r$ symbols to each copy such that the resulting projection in every copy is accepted by the local gadget. Taking the nondeterministic union of the corresponding product NFAs yields a global NFA. Amplifying a $200$-state gadget for $L_{6,11}$ obtained from the small Witt design $S(4,5,11)$, this framework gives a $3.967^k n^{O(1)}$-size NFA. We then introduce the compose-and-compress technique, which deletes the expensive middle layers of these products and replaces paths across the deleted bands with sound one-symbol shortcut transitions. We apply it twice, once for enhancing the amplification framework and again for the local gadget, obtaining the stated result.

In the proper variant of the classical hat-guessing game on a graph, an adversary properly colors the vertices from a palette of $q$ colors. Each vertex sees only its neighbors' colors and all vertices simultaneously guess their own color. The players win if at least one guess is correct. We study this game on the book graph $B_{k,n}=K_k\vee\overline{K_n}$, with $k$ mutually adjacent spine vertices and $n$ independent pages. We first give a coverability characterization valid for every fixed spine size. Write $\operatorname{HGP}(G)$ for the proper hat-guessing number of $G$. For a finite configuration $P$, let $\operatorname{supp}(P)$ denote the set of colors appearing in its tuples. Let $C_k$ be the minimum of $|P|+|\operatorname{supp}(P)|$ over all non-coverable finite configurations $P$ of proper $k$-tuples. We prove $\sup_{n\geq 1}\operatorname{HGP}(B_{k,n})=C_k$ and that $\operatorname{HGP}(B_{k,n})=C_k$ for all sufficiently large $n$. Thus, the asymptotic problem for every fixed $k$ reduces to a finite extremal invariant. In particular, coverability of two-spine configurations is equivalent to pseudoforestness, and we determine the associated extremal problem exactly: $C_2=11$, with precisely two types of extremal obstruction. Consequently, $\operatorname{HGP}(B_{2,n})\leq 11$ for every $n$, with equality for all sufficiently large $n$; we give an explicit probabilistic estimate with a stabilization threshold of at most $4\times 10^8$. We also resolve the first previously open finite cases. An explicit seven-color construction with affine symmetry proves $\operatorname{HGP}(B_{2,3})=7$. A counting-rigidity argument establishes the linear upper bound $\operatorname{HGP}(B_{2,n})\leq n+3$ for all $n\geq 4$, which together with monotonicity yields $\operatorname{HGP}(B_{2,4})=7$. Finally, a general box obstruction gives explicit uniform bounds on $C_k$.

In this paper, we consider the problem of updating the directed minimum spanning tree (DMST), when the given sample tree is subject to the weight changes, edge deletions and edge insertions. We present an implementation for updating the tree to a DMST using the weighted matroid intersection algorithm. Our algorithm focuses on maintaining a dynamic auxiliary graph, which plays a central role in the matroid intersection algorithm, and governs the iterations from the given tree to a DMST. Each iteration is guaranteed to yield an improved solution. We also provide an implementation of this algorithm and some experimental analysis.

Recently, Samanta et al. introduced a unifying perspective to common graph-associated matrices through the matrix $B_{\alpha}(G)=\alpha A(G)+(1-\alpha)L(G)$ where $\alpha\in [0,1]$. We resolve an open problem regarding the positive semi-definiteness of $B_{\alpha}$ and similar matrices. We also explore eigenvector properties of $B_{\alpha}$ through edge and vertex removal. As a consequence, we provide a basis for results on the common graph-associated matrices.

Let \[ \Bbbk[\Bbbk^3\otimes\Bbbk^2\otimes\Bbbk^2]^{U_3\times U_2\times U_2}. \] We construct an ordinary cluster family with Markov principal part. At $\zeta=-1$, the intersection of its initial Laurent ring with the three adjacent Laurent rings equals $\mathscr U$, and \[ \mathscr U=\mathcal M_u[u_\Delta], \] where $\mathcal M_u$ is the middle algebra. Its theta functions are indexed by a cone with a sixteen-element Hilbert basis. Multiplication by $u_\Delta$ pairs the Hilbert generators of mutable degrees $1$ and $-1$ and reduces each triple-weight space to the slice $\ell=\ell_0$. Counting the lattice points in this slice gives a finite sum with nonnegative summands. Determinant reduction extends the formula to all $n\times2\times2$ Kronecker coefficients.

A $(1, 1, 2, k)$-packing coloring of a graph $G$ is a partition of $V(G)$ into two independent sets, a 2-packing, and a $k$-packing. Recently, the question was posed in [A short proof that every claw-free cubic graph is (1, 1, 2, 2)-packing colorable, arXiv:2512.24001v1] as to whether every claw-free cubic graph is $(1, 1, 2, 3)$-packing colorable. We provide an answer in the affirmative in the case that $G$ is a diamond-free, claw-free cubic graph.

A dominating set in a graph $G$ is a subset $S$ of its vertices such that each vertex in $G$ is either in $S$ or adjacent to a vertex in $S$. Nordhaus-Gaddum inequalities relate the values of a graph parameter on a graph and its complement. In this setting, Keough and Shane conjecture that any graph $G$ on $n$ vertices satisfies $\partial(G) + \partial(\bar{G}) \leq 2(2^{\lfloor n/2 \rfloor} - 1)(2^{\lceil n/2 \rceil} - 1) + 2$, where $\partial(G)$ is the number of dominating sets in $G$. We partially resolve this conjecture for the bipartite case by proving the stronger bound: for a bipartite graph $G$ with nonempty bipartition $(A,B)$, it holds that $\partial(G) + \partial(\bar{G}) \leq 2(2^{|A|} - 1)(2^{|B|} - 1) + 2$. We also characterize the bipartite graphs for which equality holds.

For a positive integer $d\geq 1$, we consider the sequences $(A_{n}^{(d)})_n$ and $(x_{n}^{(d)})_n$ given by $$ A_{n}^{(d)} =\sum_{n_1+\dots+n_d=n} \frac{(2n)!}{(n_1!)^2 (n_2!)^2 \dots (n_d!)^2} \quad \text{ and } \quad x_{n}^{(d)} = \frac{A_{n}^{(d)}}{\binom{2n}{n}}. $$ They have rich combinatorial interpretations, but we focus on the analytical properties of their generating functions $A_d$ and $F_d$. We use a modified Borel transform, and algebraic and combinatorial considerations to prove that $F_d$ is annihilated by an irreducible Fuchsian differentiable operator $L_{d-1,F}$ of order $d-1$. We determine the structure of $F_d$ as a global analytic function (analytic continuations from the original disk of definition, branches, finite singularities, and the structure of $F_d$ near the finite singularities). Additionally, we show that the sequence $(x_n^{(d)})_n$ satisfies a minimal recurrence of width $r=\lfloor (d+1)/2 \rfloor$ with polynomial coefficients $$ Q_r(n+r)\,x_{n+r}+\cdots + Q_0(n)\,x_n=0, \; n \ge 0. $$ These polynomials are shown to have very specific symmetries and we compute explicitly $Q_0$, $Q_1$, and $Q_r$. Similar results about the functions $A_d$ are obtained.

The homomorphism threshold $\delta_{\text{hom}}(H)$ of a graph $H$ asks how large the minimum degree of an $H$-free graph has to be in order to force a homomorphism to a bounded $H$-free graph. Determining this threshold is in general very difficult, and the odd cycles $C_{2k-1}$ are among the most important open cases for $k\ge 3$. Ebsen and Schacht proved the general upper bound $\delta_{\text{hom}}(C_{2k-1})\le\frac{1}{2k-1}$, while a breakthrough of Sankar, using topological methods and a graph-theoretic analogue of homotopy equivalence, gave the first positive lower bound. The value $\frac{1}{2k-1}$ appeared particularly compelling: Ebsen and Schacht obtained the same exact threshold when all odd cycles of length at most $2k-1$ are forbidden, Huang, Liu, Rong and Xu later proved that it is the exact blowup threshold of $C_{2k-1}$, and Letzter and Snyder also explicitly asked whether $\delta_{\text{hom}}(C_{5})=\frac{1}{5}$. Surprisingly, we show that the upper bound can be improved. More precisely, for every integer $k\ge3$, we prove $$ \frac{1}{2\left((k-1)^{4k-5}(2k-1)+\frac{(k-1)^{4k-5}-1}{k-2}\right)} \le \delta_{\text{hom}}(C_{2k-1}) \le \frac{4(k-1)}{4(k-1)(2k-1)+1} < \frac{1}{2k-1}. $$ The new lower bound comes from a new graph-theoretic construction based on a sparse homomorphism theorem of Nešetřil and Zhu, and it improves Sankar's quantitative bound. The improved upper bound follows from a new structural argument that controls common neighborhoods along short odd paths. Our results have various consequences, in particular, every odd cycle of length at least five has pairwise distinct chromatic, homomorphism, polynomial removal, and linear removal thresholds, resolving two conjectures of Fox and Wigderson.

Let \(\cthreshold(n)\) be the largest integer \(q\) such that every loopless digraph on \(n\) vertices with at most \(q\) arcs is isomorphic to a spanning subdigraph of a self-complementary digraph of order \(n\). We prove that \(\cthreshold(6)=7\). The upper bound is witnessed by \[ \bK{3}\dunion (x\longrightarrow y\longrightarrow z), \] and follows from a direct argument with a self-complementing permutation. We also determine the complete eight-arc obstruction layer: it consists of five isomorphism classes, or three after converse digraphs are identified. All five are arc-minimal. Each nevertheless packs with an isomorphic copy of itself, so ordinary packing is strictly weaker than same-order self-complementary completion already at this first failure layer.

In a graph $G=(V,E)$, let $S$ be a non empty subset of $V$. For any $x\in S$, the vertex $x$ and its neighbors inside $S$ are defenders of $x$, whereas those lying outside $S$ are attackers on $x$. An attack on $S$ is an $|S|$-tuple of pairwise disjoint sets of attackers on vertices of $S$, whereas a defense is that of defenders. An attack on $S$ is defendable if $S$ has a defense, which provides at least as many defenders as attackers for each of its vertices. The set $S$ is a secure set if every attack on $S$ is defendable. Further, $S$ is an ultra secure set if $S$ has a defense which successfully defends $S$ against every attack. For an integer $k\geq 0$, a $k$-secure set $S$ is a secure set in which for any attack on $S$, there is a defense of $S$ with $k$ additional defenders for every vertex of $S$ having neighbors outside $S$. As a generalization of ultra security, in this paper ultra $k$-secure sets and ultra $k$-security number of a graph are introduced. A characterization of ultra $k$-secure sets is obtained and using which ultra $k$-security numbers of complete multipartite graphs and grid-like graphs are computed.

In 1943, Hadwiger formulated his celebrated conjecture, connecting the chromatic number $\chi(G)$ of a finite, simple, undirected graph with the cardinality of the largest complete minor, $\eta(G)$. The disjoint union of all finite complete graphs shows that Hadwiger's conjecture fails for infinite, but a slightly weaker version is true in these graphs, and open for finite graphs. In this note we generalize that weaker version to hypergraphs and provide a simple, general, and purely set-theoretical formulation of Hadwiger's conjecture.

For $n$ points in the plane and an $\varepsilon>0$, we construct a $(1+\varepsilon)$-spanner with $O(n/\varepsilon)$ edges in which every edge has $\tilde{O}(1/\varepsilon^3)$ crossings, hence the total number of crossings is $\tilde{O}(n/\varepsilon^4)$, furthermore the ratio between the lengths of any two crossing edges is $O(1/\varepsilon^2)$. Our spanner construction substantially improves on the previous upper bound for the number of crossings in a $(1+\varepsilon)$-spanner, and it is the first spanner construction that ensures $O(1)$ crossings per edge for any constant $\varepsilon>0$. In contrast, we construct: $n$ points in the plane for which every $(1+\varepsilon)$-spanner has $\Omega(n/\varepsilon^3)$ crossings, $n$ points for which every $(1+\varepsilon)$-spanner has an edge with $\Omega(1/\varepsilon^{5/2})$ crossings, and 4 points for which every $(1+\varepsilon)$-spanner contains two crossing edges where one is $\Omega(1/\varepsilon)$ times longer than the other.

We introduce a robust OR polynomial framework for composing positive semidefinite certificates across OR constraints. We demonstrate the power of this method in two applications. The first is on acute-free families. A set $\mathcal F=\{(x_i^{(1)},\ldots,x_i^{(r)})\}_{i=1}^M \subseteq (S^{n-1})^r$ is $r$-way acute-free if, for every $i\neq j$, there is a coordinate $t\in[r]$ such that $\langle x_i^{(t)},x_j^{(t)}\rangle\leq 0$. We write $M_r(n)$ for the maximum size of such a set, and $M_r^{\pm}(n)$ for the hypercube restriction. On the hypercube, $r$-way acute-free sets are independent sets for some strong power graph $G_n^{\boxtimes r}$. The Lovász theta number $\vartheta(G_n)$ is multiplicative but exponentially loose, whereas the Schrijver number $\vartheta'(G_n)$ gives the correct order, but is not multiplicative. We bypass this obstruction by proving a general quasi-tensorization result for the Schrijver number. That is, for every collection of graphs $G_1,\ldots, G_r$ satisfying $\vartheta'(G_i)\geq 2$, there is an absolute constant $C$ such that $\vartheta'(G_1\boxtimes \cdots \boxtimes G_r) \leq \prod_{i=1}^r \vartheta'(G_i)^{C\log r \log \vartheta'(G_i)}$. Applying this result gives that $M_r^{\pm}(n) \le M_r(n) \le (2n)^{C_0 r\log r\log(2n)}$ for some absolute constant $C_0$. The second application is on multicolor Ramsey numbers. The $r$-color Ramsey number $R_r(k)$ is the minimum $n$ such that every $r$-coloring of the edges of the complete graph on $n$ vertices contains a monochromatic copy of $K_k$. In a breakthrough result, Balister et al. [arXiv:2410.17197] showed that $R_r(k)\le \exp(-\Omega(k/r^{12}))r^{rk}$ via a geometric lemma. By improving the $r$ dependency in their geometric lemma via the OR polynomial framework, we prove that $R_r(k)\le \exp(-\Omega(k/(r^9(\log r)^6)))r^{rk}$.

We prove that the vertices of any $n$-vertex graph can be partitioned into pieces of radius $r = O(\log n)$ such that the sum of the sizes of their closed neighborhoods is at most $4n$. This answers a recent question of Bukh and Dubroff and directly yields an improvement to their upper bound on the optimal cover time of the $\epsilon$-steered random walk. We also demonstrate that our bound on $r$ is best possible up to a constant factor for graphs with strong vertex expansion.

We define a family of finite directed graphs $G(k,c,w)$ on vertex set ${\mathbb{Z}/k\mathbb{Z}}\times {\mathbb{Z}/k\mathbb{Z}}$, where $w\in{\mathbb{Z}/k\mathbb{Z}}^\times$ has multiplicative order $c$, which divides $k$. We show that each $G(k,c,w)$ is a central digraph (its adjacency matrix squares to the all-ones matrix). We deduce explicit descriptions of the graph automorphism groups for graphs of type $G(k,1,1)$; $G(k,2,-1)$ when $k$ is even; and $G(p^2, p, 1+p)$, when $p$ is prime.

An antidirected path is an oriented path in which every vertex sees either just incoming or just outgoing edges. We prove that every oriented graph with minimum semidegree at least $k$ contains an antidirected path of length $2 k -1$. This confirms a conjecture of Stein.

The cycle double cover conjecture of Szekeres and Seymour, the proof of which was recently announced by OpenAI, states that every bridgeless graph has a collection of cycles covering every edge exactly twice. We study the counting version of this statement for cubic graphs, where we count circuit double covers — collections of circuits (connected 2-regular subgraphs) covering every edge twice. We show that every 2-edge-connected 3-edge-colorable cubic graph on $n$ vertices has at least $2^{n/2-1}$ circuit double covers, matching our previously conjectured general lower bound. For every 3-edge-connected cubic graph with girth at least 16 we show a weaker exponential lower bound on circuit double covers. For both of these results we use the same system of linear equations used by OpenAI in their proof, however, we provide additional combinatorial interpretation. We characterize planarity of a cubic graph by solvability of this system of equations for arbitrary nowhere-zero $\mathbb Z_2^k$-flow. We give a condition on the flow that is equivalent to existence of a 5-cycle double cover.

We prove that at most six pairwise non-overlapping infinite unit cylinders can simultaneously touch a unit ball. Six is achievable, so the answer to Kuperberg's question is exactly six. The proof is computer-assisted in the following sense: the statement is reduced to a finite case analysis with 2,954,984 cases, and in each case a program verifies that one of three inequalities between explicit polynomials with rational coefficients holds throughout a box. Each case is an elementary arithmetic check, and the complete list of cases is provided with the paper.