Papers

A classical theorem of Frucht states that every finite group occurs as the automorphism group of a finite graph. We prove an embedded analogue for regular graphs of arbitrary degree. In particular, we show that for every $d\geq 3$ and every finite group $G$, there exists a $d$-regular graph $\Gamma$ with a strong embedding $\beta$ such that $\mathrm{Aut}(\Gamma) \cong \mathrm{Aut}(\beta(\Gamma)) \cong G.$ Further, we prove that for every such $d$ and $G$ there exists a sequence of $d$-regular graphs with corresponding strong embeddings whose genera form an unbounded sequence and whose automorphism groups are isomorphic to $G$. Along the way, we identify an oversight in Sabidussi's classical construction of regular graphs with prescribed automorphism group. We give an alternative construction that corrects this issue and strengthens Sabidussi's result by producing an automorphism group-invariant proper $d$-edge-colouring.

A simple undirected graph $M$ is called a discrete $d$-pseudomanifold if, for every vertex $v$, the induced subgraph $N_M(v)$ on the neighbors of $v$ is a discrete $(d-1)$-pseudomanifold, where a discrete $1$-pseudomanifold is defined to be an $n$-cycle with $n\geq 4$. These objects arise naturally as graph-theoretic analogues of simplicial pseudomanifolds and provide a purely combinatorial framework for studying manifold-like structures through local neighborhood conditions. Understanding discrete pseudomanifolds with a small number of vertices is therefore a fundamental problem in combinatorial topology and extremal graph theory. In this article, we first prove that every discrete $d$-pseudomanifold has at least $2(d+1)$ vertices. We then provide a complete classification of discrete $d$-pseudomanifolds with at most $2d+6$ vertices by determining all possible combinatorial types of such pseudomanifolds. Further, we establish an equivalence between discrete $d$-pseudomanifolds and edge graphs of flag normal $d$-pseudomanifolds. As a consequence, we derive a purely combinatorial characterization of flag normal $d$-pseudomanifolds with at most $2d+6$ vertices and prove that each such complex is a simplicial $d$-sphere. Finally, we show that this sphere characterization is optimal within the class of flag normal $d$-pseudomanifolds by constructing examples on $2d+7$ vertices that are not spheres. Specifically, we prove that, for $d\geq 3$, every flag normal $d$-pseudomanifold with at most $2d+7$ vertices is either a simplicial $d$-sphere or a flag triangulation of the $(d-2)$-fold suspension of $\mathbb{RP}^{2}$.

We study $H$-packings in plane triangulations for the three-vertex graphs $H\in\{P_3,K_3,P_2\cup P_1\}$. For a graph $H$, let $\lambda_H(G)$ denote the maximum size of an $H$-packing in $G$, with the convention that for $H=P_2\cup P_1$ the copies are required to be induced. For $P_3$-packings, we prove that every triangulation $G$ on $n$ vertices satisfies $\lambda_{P_3}(G)\ge \left\lfloor \frac n5\right\rfloor$, and show that this lower bound is asymptotically tight. We also study triangle packings in triangulations and provide lower bounds for $\lambda_{K_3}(G)$ in terms of the maximum degree and the degree sequence. We give a face-path characterization of triangle factors in $4$-connected plane triangulations using a hamiltonian cycle and the weak duals of the two associated maximal outerplanar graphs. Finally, for induced packings by $P_2\cup P_1$, we prove that every plane triangulation $T$ on $n$ vertices satisfies $\lambda_{P_2\cup P_1}(T)\ge \left\lfloor \frac n3\right\rfloor-2$, and show that such a packing can be found in polynomial time.

We investigate a variant of Nim called Halve Nim, which in addition to the standard moves of Nim, we allow replacing each pile of coins with half its amount. We determine the P-positions of all two-pile games of Halve Nim. Also, we determine the P-positions of all three-pile games of Halve Nim in which one pile has at most ten coins.

We determine the asymptotic maximum number of unlabelled copies of $P_{2r+1}$ in graphs with prescribed edge density, where $r\ge1$ is fixed and $P_{2r+1}$ denotes the path on $2r+1$ vertices. If an $n$ vertex graph $G$ has edge density $c=2e(G)/n^2$, then the maximum is $\frac12S_r(c)n^{2r+1}+O(n^{2r})$ for $0<c\le c_r$, and $\frac12c^{r+1/2}n^{2r+1}+O(n^{2r})$ for $c_r\le c<1$, where $S_r(c)$ is the value given by the quasi-star construction and $c_r\in(0,1)$ is an explicit algebraic transition point. Thus the quasi-star construction is asymptotically extremal below the transition, while the quasi-clique construction is asymptotically extremal above the transition. This extends the quasi-star versus quasi-clique theorem of Ahlswede and Katona for $P_3$ and the theorem of Nagy for $P_5$ to all paths with an odd number of vertices. The proof reduces the problem to threshold graphons and then to two endpoint families. The three-step endpoint is handled by reducing the required inequality to coefficient nonnegativity in a Bernstein expansion, which is proved by a direct combinatorial argument.

We prove that if $f\in \mathbb Z[x]$ is a monic polynomial of degree $k\geq 2$, then there exists a constant $c>0$, depending only on $f$, and finite sets $A\subset \mathbb R$ of arbitrarily large size such that \[ |f(A)|\leq |A|^{k-c}, \] where $f(A)$ is interpreted in the Minkowski sum-product sense. In particular, taking $f(x)=x^2+x$, this gives a power-saving upper bound for $AA+A$, answering a question raised by Roche-Newton, Ruzsa, Shen, and Shkredov.

We study the siblings version of the coupon collector problem. A main collector stops when every coupon type has appeared at least once, duplicates are passed successively to later siblings, and $U_j^N$ denotes the number of empty spaces in collector $j$'s album at the main completion time. We prove three results. First, for every fixed $N$ and $j\ge2$, $\E U_j^N$ is uniquely maximized over positive coupon distributions by the uniform distribution; in fact it decreases strictly along every nonconstant ray from the uniform vector. Second, in the uniform model, $U_j^N$ is stochastically increasing in $N$, and we construct an increasing coupling using top spacings of exponential order statistics. Third, for fixed album indices $2,\ldots,J$, the naturally normalized vector converges jointly to $(W,\ldots,W)$, where $W$ is exponential with mean one. We also derive exact Poissonized and alternating-subset formulae and give a transfer principle for leading expectation asymptotics.

The paper studies the graph algebra whose monomial basis is naturally indexed by simple graphs on a fixed set of vertices. This algebra is at the same time the algebra of pseudo-Boolean functions on the Boolean cube and a natural object of algebraic combinatorics, related to the Boolean lattice of subsets of the edge set of the complete graph. The main aim of the paper is to study two compatible representation-theoretic structures on this algebra: the action of the Lie algebra $\mathfrak{sl}_2$, arising from the operators of adding and deleting one edge, and the action of the pair group $S_n^{(2)}$, induced by the renumbering of vertices. It is proved that the graph algebra with this $\mathfrak{sl}_2$-action is isomorphic to a tensor power of the standard two-dimensional $\mathfrak{sl}_2$-module, and on this basis its decomposition into irreducible $\mathfrak{sl}_2$-modules is obtained. Primitive spaces, that is, the kernels of the edge-deletion operator on rank components, are also described, and it is shown that they have a natural interpretation in terms of two-row Specht modules. It is then established that the $\mathfrak{sl}_2$-action commutes with the action of the pair group. It follows that the space of graph invariants also inherits the structure of an $\mathfrak{sl}_2$-module. Using Schur–Weyl duality, primitive invariants are described through the fixed parts of the restrictions of two-row Specht modules from the full symmetric group on the edge set to the pair group. As a consequence, the classical enumeration of non-isomorphic graphs by the number of edges receives a representation-theoretic refinement: the orbital components entering the Burnside–Polya formula decompose into natural primitive contributions associated with the $\mathfrak{sl}_2$-structure and two-row Specht modules.

Automated conjecturing programs scan collections of graphs for inequalities between invariants that no stored graph violates, then offer the survivors for proof or refutation. TxGraffiti, one such program, conjectured that every nontrivial connected graph $G$ satisfies $\alpha(G) \ge \bigl(a(G) + R(G)\bigr)/\Delta(G)$, where $\alpha$ is the independence number, $a$ the annihilation number, $R$ the residue, and $\Delta$ the maximum degree. Established only for two special families of graphs, the conjecture has otherwise remained open. The note proves the degree-sequence inequality $a \le \tfrac{\Delta+1}{2}W$, where $W$ is the Caro-Wei sum; the same inequality is known for the independence number in place of $a$. Combined with the classical lower bounds $\alpha \ge R$ and $\alpha \ge W$, it proves the conjecture for every connected graph of maximum degree at least three, and a direct argument settles maximum degree two; the conjecture fails only for the single edge, of maximum degree one. The inequality also brackets the independence number between the polynomial-time quantities $R$ and $a$, within a factor $(\Delta+1)/2$. The conjecture's bound is sharp, with equality attained, for instance, by the complete graph on four vertices.

We study the $p$-colored top-$m$-to-random shuffle on $C_p\wr S_n$ when the block size $m=m_n$ grows with $n$. Let $E_{k_n}^{(m_n)}$ be the number of labels never touched after $k_n$ independent uniform $m_n$-subset draws, and set $b_n=n-m_n$, $q_n=b_n/n$, and $\lambda_n=nq_n^{k_n}$. We prove that if $\lambda_n\to\lambda\in(0,\infty)$ and $b_n\to\infty$, then $E_{k_n}^{(m_n)}\Rightarrow\mathrm{Poisson}(\lambda)$. Combining this with the exact nested-set reduction for colored top-$m$-to-random shuffles, we obtain growing-block total variation, separation, and integrated likelihood-ratio profiles. In particular, if $Q_{n,p}^{(m_n)}$ is the one-step law and $U_{n,p}$ is uniform on $C_p\wr S_n$, then the separation distance from $(Q_{n,p}^{(m_n)})^{*k_n}$ to $U_{n,p}$ tends to $1-e^{-\lambda}(1+\lambda)$ for $p=1$ and to $1-e^{-\lambda}$ for $p\ge2$. The criterion applies to small blocks, proportional blocks, and near-full blocks.

The specification number $\sigma_n(f)$ of a Boolean threshold function $f$ on $n$ variables is the least number of points whose $f$-values determine $f$ uniquely among all threshold functions. Its essential points form the unique minimum such set. We develop Zuev's geometric interpretation: the threshold functions are the chambers of a central hyperplane arrangement in the $(n+1)$-dimensional space of weights and thresholds, and the essential points of a function correspond exactly to the facets of its chamber, so the specification number is the chamber's facet number. The lower bound $\sigma_n(f)\ge n+1$ becomes the fact that a pointed full-dimensional cone has at least $n+1$ facets, with equality for simplicial chambers. The average specification number $\overline\sigma_n$ becomes an average facet count. We evaluate this average exactly via the resonance arrangement and bound it through a theorem of Fukuda, Tamura, and Tokuyama, obtaining $\overline\sigma_n\le 2n$; hence $\overline\sigma_n=\Theta(n)$. This settles a question of Gutekunst, Mészáros, and Petersen. The method also extends to polynomial threshold functions. The same geometry links threshold functions with a threshold zonotope, whose vertices are modified Chow vectors. Its one-skeleton is the one-inclusion graph, and a vertex's degree is the specification number of that function. Finally, we treat the operations of Lozin et al. on functions of minimum specification number. Adding a variable and extending on a variable both take the product of a chamber closure with a half-line, preserving simpliciality. For the symmetric-variables extension we give an exact thresholdness criterion and show that minimum specification number is preserved whenever the extension is a threshold function. We also resolve a question they pose concerning a fourth operation.

The KKL Theorem, a seminal result in boolean function analysis, characterizes the structure of low-influence (non-expanding) functions on the hypercube. While recent years have seen breakthrough results across a variety of areas relying on analogs of the KKL Theorem beyond the cube (e.g., on product spaces, Grassmann graphs), further progress has been inhibited by our poor understanding of the phenomenon across more general domains. Motivated in this context, Bafna, Hopkins, Kaufman, and Lovett (STOC 2022) and Gur, Lifshitz, and Liu (STOC 2022) proved a generalized KKL-type Theorem for spectral high dimensional expanders (HDX). Their results, however, remain highly restricted due to strong quantitative expansion requirements on the underlying complex. In this work, we introduce a simple local-to-global method for analyzing low influence functions on simplicial complexes. Using this method we prove a local-to-global KKL-type Theorem: any simplicial complex whose links satisfy a KKL-Theorem also satisfies such a result globally. Building on Gotlib and Kaufman (RANDOM 2023), we also prove a weaker dimension-dependent KKL-type Theorem for simplicial complexes with any non-trivial (two-sided) expansion. As concrete applications of our framework, we give the first characterization of non-expanding functions on `combinatorial' HDX such as dense clique complexes and a corresponding Kruskal-Katona Theorem, as well as a small-set expansion theorem for the Ramanujan Complexes of Lubotzky, Samuels, and Vishne (EJC '05).

Let $\mathcal{P}$ be a set of $n$ points in $\mathbb{R}^2$, with a convex hull of size $O(n/\log n)$. We prove that $\Omega(12.24^n)$ plane graphs can be drawn on $\mathcal{P}$, the first non-trivial bound for this problem. We also show that a random plane graph, uniformly chosen from the set of all plane graphs of $\mathcal{P}$, has at most $n/12.24$ isolated vertices. This improves upon a previous bound of $n/10.18$. Our analysis is based on studying the expected vertex potentials in a random plane graph. The potential of a vertex is its degree plus the number of vertices visible from it. We show that this quantity can be used to study numbers of plane graphs.

A graph homomorphism is an integer-valued function on the vertex set of a graph that assigns values differing by exactly one to adjacent vertices. We consider uniformly random homomorphisms on general finite trees, conditioned to take the value zero at all leaves, and study the distribution of the value at the root. Our main result is a stochastic comparison, both from above and below, between the absolute values of the homomorphism value at the root and certain discrete Gaussian-like random variables. In particular, we obtain a subgaussian tail bound valid for all deviations, a matching lower bound that holds up to a certain threshold, and upper and lower variance bounds that differ by a constant factor. These bounds depend solely on the effective resistance between the root and the leaves in the associated electrical network. As a consequence, in the setting of infinite locally finite trees, we obtain that the homomorphism model is localized on transient trees and delocalized on recurrent trees. Analogous results are obtained for random integer-valued Lipschitz functions. Our results extend previous results of Benjamini–Häggström–Mossel on homomorphisms on regular trees, of Peled–Samotij–Yehudayoff on Lipschitz functions on regular trees, and of Lammers–Toninelli on homomorphisms on trees of minimum degree at least three.

This paper presents reproducible experimental evidence on unit-distance graph density. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures, used in the literature for generating unit-distance graphs. When applied to large unit-distance graphs, our method generates large graphs with a scaling exponent of 1.17, that surpasses recent theoretical predictions on graph density growth.

We study standard spherical growth rates of right-angled Coxeter groups through the clique polynomial of the defining graph. We prove that every even degree at least four occurs as the degree of a strongly primitive Salem growth rate: for each $d \geq 2$, there are infinitely many connected $K_{2d+1}$-free defining graphs whose full reciprocal-radius polynomial is an irreducible Salem polynomial of degree $2d$. We also prove independence-polynomial obstructions for prescribed Salem polynomials, including a sharp first-coefficient bound $a_1 \leq -5$, and apply them to Lehmer's polynomial and its suspension multiples.

Let $N_{\alpha}(d)$ denote the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle $\arccos(\alpha)$. Balla conjectured that, if the spectral radius order $\kappa_{\frac{1-\alpha}{2\alpha}}$ of $\frac{1-\alpha}{2\alpha}$ is finite, then $$N_{\alpha}(d)\leq \max\left\{\frac{(1-\alpha^2)(1-2\alpha^2)}{2\alpha^4},\left\lfloor\frac{\kappa_{\frac{1-\alpha}{2\alpha}}(d-1)}{\kappa_{\frac{1-\alpha}{2\alpha}}-1}\right\rfloor\right\},$$ for any $d\geq 1$. The conjecture has previously been verified only for $\alpha\in\left\{\frac{1}{3},\frac{1}{5},\frac{1}{1+2\sqrt{2}}\right\}$. In this paper, we prove that this conjecture holds for $\alpha=\frac{1}{1+2\sqrt{3}}$ and $\alpha=\sqrt{5}-2$. On the other hand, we show that Balla's conjecture fails for infinitely many $\alpha$.

This dissertation develops a framework for embedding arbitrary connected graphs isometrically into Cayley graphs of abelian groups, with applications to harmonic analysis on networks. It addresses representing irregular graph-structured data within highly symmetric algebraic hosts, on which classical Fourier theory applies verbatim rather than by analogy. The theoretical core is twofold. First, we introduce edge relations phi, Phi, and Psi that detect metric parallelism, a strict generalization of the Djokovic-Winkler relation beyond bipartite and partial-cube structures, with a transitive prune operation converting them into candidate same-generator edge partitions. Second, we prove the Cocycle/Quotient Labeling Theorem: any edge partition induces a most-generic consistent vertex labeling as a GF(2) quotient of dimension k = t - rank(A), where A is the cycle-class parity matrix; the labeling can fail only by shortcuts, never by stretching. With a shortcut-repair loop terminating in the isometric spanning-tree embedding, this gives a universal algorithm: every connected graph G embeds isometrically into a Cayley graph of (Z_2)^k with k <= n-1, verified exhaustively on all 995 connected graphs of at most seven vertices. A bounds theory follows: k >= max(diam(G), ceil(log2 n)); stars satisfy k_min(K_{1,q}) = ceil(log2 q) + 1, exponentially below the naive dimension; odd cycles require k = n-1. We then generalize the quotient machinery from GF(2) to Z via the Smith Normal Form, giving embeddings into products of cyclic groups. The primary application is harmonic analysis: these embeddings ground Fourier analysis, convolution, and wavelet transforms on graph signals, preserving translation-modulation duality, convolution theorems, and Plancherel identities that matrix-based graph signal processing lacks. We name this framework Group-Embedding-based Graph Signal Processing (GE-GSP).

For integers $1\le r\le n+1$, let $N(n,r)$ denote the least number of chains in the Boolean lattice $B_n=2^{[n]}$ that cover every strict $r$-term chain. The case $r=1$ is the classical chain-decomposition problem and is generalizing Dilworth's theorem and Sperner's theorem. We study two complementary regimes. First, when $r>1$ is fixed and $n\to\infty$. Let $$M(n,r):= \max_{\substack{ a_0+\cdots+a_r=n a_0,a_r\ge 0,\ a_i\ge 1\ (1\le i\le r-1) }} \binom{n}{a_0,\ldots,a_r}.$$ We prove that lower and upper bounds which differ only by a logarithmic factor: \[ M(n,r)\le N(n,r)\le \left(\frac r2+o(1)\right)\log n\cdot M(n,r). \] Second, we consider the near-maximal regime $N(n,n-t)$, where $t>0$ is fixed. We prove that \[ N(n,n-t)= (\gamma_{t-1}+o(1))n!, \] where $\gamma_d$ is the limit of the density of a minimum size subset of the hypercube $Q_n$ that meets all $d$-dimensional subcube.

Let $f_3(N)$ be the least integer such that every set $A\subseteq\{1,\ldots,N\}$ of size at least $f_3(N)$ contains distinct elements $a,b,c\in A$ such that $a+b\in A$, $a+c\in A$, and $b+c\in A$. We prove that $f_3(N)\le 5N/8+O(1)$. Together with the standard construction $[N/8,N/4]\cup[N/2,N]$, this gives $f_3(N)=5N/8+O(1)$, resolving Erdős Problem 865. The proof is self-contained. An earlier conditional version of the reduction has also been formalized in Lean 4/Mathlib with no sorries and no added axioms.

Cycle rank is a depth parameter for digraphs introduced by Eggan in 1963. Gruber (DMTCS 2012) and Giannopoulou, Hunter, and Thilikos (DAM 2012) asked whether the problem of determining if a given digraph has cycle rank at most $w$ is fixed-parameter tractable parameterized by $w$. We provide such algorithms for semi-complete digraphs, and for digraphs of bounded directed clique-width. Specifically, we show that given an $n$-vertex semi-complete digraph~$G$ and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^{w})} n^2$ determine whether $G$ has cycle rank at most~$w$. The proof is reduced to the case of bounded directed clique-width, and we then show that given an $n$-vertex digraph $G$ with a directed clique-width $k$-expression and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^k)}n$ determine whether $G$ has cycle rank at most $w$. Additionally, we consider the Minimum Feedback Arc Set problem on semi-complete digraphs, and show that it can be solved in time $n^{\mathcal{O}(w)}$, when a cycle rank decomposition of depth $w\ge 1$ is given.

We introduce a binary matroid framework for the enumeration of mod $2$ toric-colorable seeds of fixed Picard number. Working with binary matroids up to isomorphism, we organize them through their contraction structure and recursively enumerate weak pseudomanifold subcomplexes by a dynamic programming algorithm. The resulting method combines these structural reductions with a Gray-code traversal of the mod $2$ kernel of the ridge–facet incidence matrix. Using this framework, we complete the classification of mod $2$ toric-colorable seeds of dimension four and Picard number five, proving that there are exactly $198{,}846$ isomorphism classes. This case was computationally infeasible for the GPU-based algorithm previously used by Choi, Jang, and Vallée to treat Picard number four. We further verify that each of these seeds admits an integral characteristic map. As a validation of the method, we also reproduce that Picard number four classification: the weak pseudomanifold enumeration stage drops from over ten days on a GPU to ten minutes on a single CPU.

Let $A$ be a finite non-empty subset of an abelian group $G$, and let $r_A(d)=|\{(a,a')\in A^2:a-a'=d\}|$. Croot and Lev asked whether the pointwise half-threshold condition $r_A(d)\ge |A|/2$ for every $d\in A-A$ forces $A-A$ to be either a subgroup or a union of three cosets. We resolve this open problem in its sharp general form by identifying the essential obstruction: the statement is false in arbitrary abelian groups, but becomes true after excluding non-zero two-torsion. More precisely, if $G$ is two-torsion-free and the half-threshold condition holds, then either $A-A$ is a finite subgroup of $G$, or there are a finite subgroup $H\le G$ and elements $x,g\in G$ such that \[ A=(x+H)\cup(x+g+H). \] The two-torsion-free hypothesis is essential: for every $r\ge1$ we construct $A\subseteq\F_2^{2r+1}$ with $A-A=\F_2^{2r+1}\setminus\{t\}$ such that every non-zero represented difference has exactly $|A|/2$ representations, giving genuine counterexamples to the Croot–Lev conclusion. The proof of the positive result combines a Kneser quotient reduction with Lev's formulation of Kemperman's critical-pair theory.

In this note, we give a Turán theorem for Cayley graphs $\Cay(\Z_p,S)$ over prime cyclic groups $\Z_p$. For a graph $F$ and a finite abelian group $G$, define the Cayley–Turán number by \[ \exCay(F,G) = \max\{|S|:S=-S\subseteq G\setminus\{0\},\ \Cay(G,S)\text{ is }F\text{-free}\}. \] Using a polynomial method, we prove that for every odd prime $p$ and every $1\le r\le p-1$, \[ \exCay(K_{r+1},\Z_p) = p-1-2\left\lfloor\frac{p}{r+1}\right\rfloor . \] The extremal construction is the complement of the short-difference interval \[ D_0=\{0,\pm1,\ldots,\pm\lfloor p/(r+1)\rfloor\}. \] We also discuss what changes for general finite abelian groups, showing why the exact prime-cyclic formula does not extend verbatim to composite cyclic groups.

An $(n,R)$-covering sequence over a finite alphabet $\Sigma_q = \{0,1,\dots, q-1\}$ is a cyclic sequence whose consecutive length-$n$ windows form a covering code of radius $R$. Equivalently, every word in $\Sigma_q^n$ is within Hamming distance $R$ of at least one window. We give a deterministic and explicit construction of such sequences whose length, for every fixed alphabet size $q$, every fixed radius $R$, and every sufficiently large $n$, attains the sphere-covering lower bound up to a constant factor depending only on $q$ and $R$. Thus, in the fixed-radius regime, the construction removes the logarithmic factor in the general probabilistic upper bounds of [Chung and Cooper, Random Structures & Algorithms, 2004] and [Vu, Advances in Applied Mathematics, 2005]. It also complements the earlier explicit constructions of [Chee, Etzion, Ta, and Vu, Designs, Codes and Cryptography, 2025], which include constant factor bounds for the special binary radius-one families \(n=2^a-1\) and \(n=2^a\), where \(a\ge1\).

For an integer $q\ge 2$ and a graph $F$ satisfying $q\mid e(F)$, the zero-sum Ramsey number $R(F,\mathbb Z_q)$ is the least integer $n$ such that every edge-labeling $w\colon E(K_n)\to \mathbb Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\mathbb Z_q$. Write $K_{s,t}$ for the complete bipartite graph with $s$ vertices on one side and $t$ vertices on the other side. We prove that for every $q\ge2$, there is an explicit threshold $S(q)$ such that $R(K_{s,qk},\mathbb Z_q)=s+qk$ for all $s\ge S(q)$ and all $k\ge1$. We also determine the zero-sum Ramsey number of $K_{s,3k}$ over $\mathbb Z_3$ for all $s\ge2$ and $k\ge1$. We prove that $R(K_{s,3k},\mathbb Z_3)=s+3k$, except when $s=2$ and $k\ge1$, or when $s\in\{3,4,5,7\}$ and $k=1$. In these exceptional cases, $R(K_{s,3k},\mathbb Z_3)=s+3k+1$. In particular, this shows that the threshold $S(q)$ is best possible for \(q=3\).

Given a graph $G$, the domination number $\gamma(G)$ is the minimum cardinality of a dominating set in $G$, and the packing number $\rho(G)$ is the maximum cardinality of a set of vertices that are pairwise at distance at least $3$. The ratio between these parameters has been widely studied in several graph classes. It is known that $\gamma(G) \le 2\rho(G)$ for claw-free subcubic graphs, up to finitely many exceptions, and that $\gamma(G) \le 32\rho(G)$ for unit disk graphs. In this paper, we improve the latter bound by showing that $\gamma(G) \le 16\rho(G)$ for a unit disk graph $G$. For the former bound, we show that it can be improved in the cubic bridgeless setting; more precisely, every bridgeless claw-free cubic graph $G$ satisfies $\gamma(G) \le \frac{7}{4}\rho(G) + \frac{5}{6}$. These results are not tight. In fact, we give example of an infinite family of bridgeless cubic graphs $G$ with $\gamma(G) = 5\rho(G)/4$ and an infnite family of unit disk graphs $G$ in which $\gamma(G) = 3\rho(G)$.

Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong Sárközy-type result: if $A\subseteq [N]$ has no nonzero difference equal to a sum of two squares, then $|A|\ll_\varepsilon N^{7/8+\varepsilon}$ for every $\varepsilon>0$, improving upon an earlier quasipolynomial bound due to Rice. The shape of this bound is optimal, as a construction of Younis yields a set $A\subseteq [N]$ with $|A|\gg N^{1/2}$ such that $(A-A)\cap S_N=\emptyset$.

The existence of a Moore graph of degree $57$ is the last open case in the Hoffman-Singleton classification of Moore graphs of diameter two. We prove that such a graph, if one exists, has no involutory automorphisms; consequently its automorphism group has odd order. The proof relies on a new trace-rank identity, valid for every finite graph and obtained from the Brauer quotient of $p$-permutation lattices: the trace of an automorphism of prime order $p$ on a spectral summand whose projection idempotent preserves $p$-adic permutation lattice equals the rank, over the residue field of characteristic $p$, of the corresponding idempotent restricted to the fixed vertices. For strongly regular graphs, this upgrades the classical character-value congruences for prime-order automorphisms to exact rank equalities. As further applications, the identity yields new restrictions on automorphisms of odd prime order of the hypothetical Moore graph, and it sharpens known automorphism analyses of other hypothetical strongly regular graphs.

We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means. All three gap-sums admit a common interpretation: each equals the gap size multiplied by the appropriate mean of the missing integers. Building on this, we prove a general sparse summation theorem expressing the sum of a strictly monotonic function over a sparse integer sequence as the full range sum corrected by the gap-sums of the missing portions. Specializing on the three Pythagorean means recovers a classical formula of al-Kāshī from the fifteenth century in the arithmetic case, and yields explicit formulas in the geometric and harmonic cases. As a concrete application of the geometric case, we derive a product identity involving the Fuss–Catalan numbers. Applying the harmonic case to the Fibonacci and Lucas sequences, we establish that the harmonic gap-sum converges to $\ln(\alpha)$ exponentially, where $\alpha$ is the golden ratio, and derive explicit two-term asymptotic expansions for the tails of the reciprocal Fibonacci and Lucas series with closed-form coefficients, and establish the asymptotic formula $H_{u_n} \sim n\ln(\alpha)$ for both $u_n = F_n$ and $u_n = L_n$, with explicit $O(1)$ error terms that differ due to their distinct initial conditions. As a further consequence, by comparing the gap-sum expansions with the classical Hardy–Wright expansion of harmonic numbers, we derive exact series identities expressing Euler's constant $\gamma$ in terms of harmonic numbers at Fibonacci and Lucas indices, and obtain a new identity relating the reciprocal Fibonacci constant $\psi$ and the reciprocal Lucas constant $\psi_L$.