Papers

In this paper, we study and characterise the natural embedding of the Ree-Tits octagons. Keywords: Ree-Tits Octagon, Metasymplectic space, generalised octagon, generalised polygon

We propose a new way to construct one block design from another, thus linking some block designs in a new way. At the same time, we find relationships between block designs and systems of pairs. In particular, these relationships allow us to generate some systems of pairs.

The random cut is one of the most fundamental shuffles in card-based cryptography: it rotates a sequence of face-down cards by a secret amount. Under this shuffle, two sequences of cards are indistinguishable if and only if they are cyclic shifts of each other. This motivates the question of whether, given two sequences of cards, inserting cards at matching positions can make them indistinguishable. A previous study shows that such an insertion is always possible when any cards may be inserted, as long as the two words are permutations of each other. This paper considers a stronger restriction: if the cards are binary, carrying only 0 or 1, can we insert only 0s to make the sequences indistinguishable? We call two words 0-cyclically equalizable if one can insert 0s into both sequences at matching positions so that the resulting words are cyclic shifts of each other. Our main result is that two binary words of equal length are 0-cyclically equalizable if and only if they have equal Hamming weight, that is, the same number of 1-bits. Since equal Hamming weight is clearly necessary, the content of the paper is to show that it is also sufficient. Our proof is constructive: we encode a pair of binary words as a single word over the four-letter alphabet {A, B, X, O}, reduce equalizability to a simpler condition in this encoding, and build the required insertion explicitly.

A permutation $\pi \in S_n$ is (32-1)-avoiding when there do not exist $1 \leq i < i+1 < j \leq n$ such that $\pi_i > \pi_{i+1} > \pi_j$. We determine the maximum inversion number for (32-1)-avoiding permutations and count the number of permutations that achieve this maximum. We then provide a direct construction that enumerates these permutations.

A family $\mathcal{F}$ of subsets of $[n] := \{1,2,\ldots, n\}$ is called maximal $k$-wise intersecting if every collection of at most $k$ members of $\mathcal{F}$ has a non-empty intersection, and adding any other set to $\mathcal{F}$ breaks this property. An old question by Erdős and Kleitman from 1974 asks for the minimum size of a maximal $k$-wise intersecting family. The case $k = 3$ is known for all sufficiently large $n$, but the problem remains open for all $k \geqslant 4$. The previous best-known upper bound is by Janzer, which has a leading term $(k-1)2^{k-3}2^{n/(k-1)}$ for sufficiently large $n$ divisible by $k-1$. In this note, we improve this bound to $(4k-10)2^{n/(k-1)}$, which reduces the dependence on $k$ in the leading coefficient from exponential to linear and is within a factor of $4$ of the known lower bound.

For a degree sequence $d$, the realization graph $\mathcal{G_F}(d)$ is the graph whose vertices are the labeled realizations of $d$, two of which are adjacent if they differ by a single $2$-switch. We prove that for every degree sequence $d$ and every realization $S$ of $d$, the graph $\mathcal{G_F}(d)$ contains a Hamilton path starting at $S$. This answers a question of Barrus (2016), which was also raised independently by Mütze (2023) in his survey of combinatorial Gray codes. As a consequence, an embedding observation of Arikati and Peled (1999) implies that for any vectors $R$ and $C$ of non-negative integers, the interchange graph $\mathcal{A}_{\mathcal F}(R,C)$ of $(0,1)$-matrices with row sums $R$ and column sums $C$ contains a Hamilton path starting at any prescribed matrix, thereby resolving a question of Brualdi (1980).

Motivated by recent results \cite{charamaras_kousek_mountakis_radic2025BBingroups} on infinite sumsets of the form $B+B=\{b_1+b_2:b_1,b_2\in B\}$ in large subsets of abelian groups, and an old problem of Owings \cite[Problem E2494]{Owing_problems} about the partition regularity of $B+B$ in $2$ colours, we show the following theorem. Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $D: g\mapsto g+g$ has finite kernel. Let also $\Phi=(\Phi_N)_{N}$ be any Folner sequence in $G$ and $\Phi/2=(D^{-1}(\Phi_N))_{N}$. Then, if $A\subset G$ is such that $d_{\Phi}(A)+d_{\Phi/2}(A)>1$, there is an infinite set $B\subset G$ and some $t\in G$ for which $t+B+B\subset A$. We prove that this result implies the main theorem in \cite{charamaras_kousek_mountakis_radic2025BBingroups}, and construct an example to show the reverse implication does not hold. Moreover, we show that our main theorem is optimal in a strong sense. Namely, for any countable abelian group $(G,+)$ with the aforementioned assumptions – which are necessary – there exists a Folner sequence $\Phi$ and a set $A\subset G$ so that $d_{\Phi}(A)+d_{\Phi/2}(A)=1$, but there is no infinite set $B\subset G$ and $t\in G$ for which $t+B+B\subset A$.

We determine the generalized Turán number of vertex-disjoint paths of length two and characterize all corresponding extremal graphs. Our proof combines the Lovász form of the Kruskal–Katona theorem with a discrete convexity argument.

We investigate the Hales-Jewett numbers and some variants of them, both computationally and via enumerative and probabilistic arguments. In particular, we give the improved lower bound formula on the combinatorial-geometric variant of the Hales-Jewett numbers. {As a consequence of one of the variants investigated, we show that the Milton Bradley game Connect Four played with 2 players on a 5-dimensional hypercube of any side length can end in a draw.

Let $X$ be a finite set of integers with cardinality $\nu = \kappa \lambda$. A modular balanced partition design is a triplet $(X, \mathcal{A}, \mathcal{B})$ satisfying the following conditions: \begin{itemize} \item $\mathcal{A}$ is a partition of $X$ into $\kappa$ blocks of size $\lambda$, such that every element of $X$ appears in exactly one block. If $\mathcal{A} = \{A_1, A_2, \ldots, A_{\kappa}\}$, then $\sum_{a\in A_i} a \equiv i \lambda \pmod{\nu}$, for each $i=1,2,\ldots,\kappa$ \item $\mathcal{B}$ is a partition of $X$ into $\lambda$ blocks of size $\kappa$, such that every element of $X$ appears in exactly one block. If $\mathcal{B} = \{B_1, B_2, \ldots, B_{\lambda}\}$, then $\sum_{b\in B_j} b \equiv j \kappa \pmod{\nu}$, for each $j=1,2,\ldots,\lambda$ \item $A_i \cap B_j$ has exactly one element, for any $A_i \in \mathcal{A}$, and $B_j \in \mathcal{B}$. \itemize} We prove the necessary conditions for the existence of a modular balanced partition design. Moreover, we investigate and identify a relationship between a modular balanced partition design and a subgroup magic rectangle. Then by using affine automorphisms of an Abelian group, we prove the existence of non-isomorphic modular balanced partition designs. Finally, we provide a method to construct a transversal design via a modular balanced partition design.

A longstanding conjecture of Seymour, called Seymour's second neighborhood conjecture, states that every oriented graph $D$ contains a vertex $x$ with $|N^{++}_D(x)|\geq |N^{+}_D(x)|$. The conjecture was verified in a few special classes of oriented graphs, and it remains open for general oriented graphs. We study a stronger property, asking for a vertex $x$ such that there exists a complete matching from $N^+_D(x)$ to $N^{++}_D(x)$. We prove that this stronger version holds for every oriented graph with minimum out-degree at most $5$, and also for every $5$-anti-transitive oriented graph. This implies that every oriented planar graph satisfies the stronger version.

Let $G$ be an $n$-vertex graph with clique number $\omega(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\frac{1}{\omega(G)}\right)n. $$ This strengthens Wilf's classical spectral Turán theorem and resolves a conjecture of Elphick and Wocjan. Adopting the relaxation of our companion paper on the square-energy conjecture, we reduce the theorem to a Motzkin–Straus inequality for doubly nonnegative matrices, which we prove via a local inverse-probability estimate for the Caro–Wei greedy algorithm on the complement.

Let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$, respectively. We prove the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$ \min\{s^+(G),s^-(G)\}\ge n-1. $$ The proof introduces a new framework for square-energy estimates, in which the Hadamard squares of positive semidefinite matrices that encode these spectral quantities are relaxed to the full doubly nonnegative cone.

A feedback vertex set of a graph is a set of vertices whose deletion leaves a forest. In 2016, Dross, Montassier, and Pinlou conjectured that every planar graph $G$ of girth at least $g$ admits a feedback vertex set of size at most $e(G)/g$. In this note, we confirm this conjecture by connecting this problem with signed graphs. The frustration index of a signed graph $(G,\Sigma)$ is defined as the minimum number of negative edges among all signatures on $G$ that are switching-equivalent to $\Sigma$. Equivalently, it is the minimum number of edges whose deletion results in a balanced subgraph of $(G,\Sigma)$. We show that the minimum size of a feedback vertex set of a planar graph is bounded above by the maximum frustration index over all signatures of the graph, and thereby provide a tight upper bound on the size of the minimum feedback vertex set, which resolves the conjecture of Dross, Montassier, and Pinlou (2016).

In an incomplete round robin tournament, teams play against different sets of opponents. Given varying strengths of the teams, this raises a fairness issue. We establish the complexity of determining whether a fair draw exists under various scenarios involving the presence of pots (i.e., strength based groups of teams), teams coming from the same association, connectivity requirements, and others. We also show experimentally how fair incomplete round robin tournaments can be generated.

This work establishes an improved lower bound for the spectral radius of a graph given its average degree. The new bound follows from an exact solution of the fractional relaxation of the problem. Our findings lead to an affirmative answer to a conjecture by Hong (1993) for graphs with specific average degrees – as the extremal graphs that meet our bound are proven to have a minimal and maximal degree that differ by at most one. Furthermore, we provide an exact characterization of the conditions that permit such discrete realizations. We prove that for a fixed number of vertices $n$, the number of valid edge configurations grows at least linearly with $n$, achieving an average asymptotic order of $\Theta(n\log n)$.

Let $R(T_n,C_m)$ denote the Ramsey number of a tree $T_n$ on $n$ vertices versus a cycle $C_m$ of length $m$. Burr, Erdős, Faudree, Rousseau, and Schelp (1982) asked for the least function $f(m)$ such that $R(T_n,C_m)=2n-1$ for every odd $m\ge 3$ whenever $n\ge f(m)$. They proved that $f(m)\le 756m^{10}$. This bound was later improved to $25m$ by Brennan (2016) and to $4m-8$ by Fan and Lin (2025). In this note, we show that $f(m)\le 2m-4$ by using a different method and conjecture that $f(m)=\lceil (2m-1)/3\rceil$.

This thesis investigates two central directions in algebraic graph theory, with an emphasis on spectral methods: spectral determination of graphs and transitivity properties of generalized-Hamming graphs and their complements. The first part focuses on graphs that are determined by the spectra of associated matrices. We study spectral determination with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, with particular emphasis on the adjacency spectrum. We survey existing results on graphs determined by their spectrum and develop new proof techniques for establishing spectral uniqueness. In particular, we present new proofs for the spectral characterization of complete bipartite graphs and Turán graphs, as well as some new results related to the spectral characterization of the important family of strongly regular graphs. In addition, we introduce a new family of graphs, called the graphs of pyramids, and prove that they are determined by their adjacency spectrum using tools from matrix analysis, such as Cauchy's interlacing theorem and Schur complements. The second part of the thesis studies generalized-Hamming graphs, a family of Cayley graphs that generalize the sub-family of Hamming graphs, and their complements. We classify the parameters for which these graphs are edge-transitive or even distance-transitive. Our analysis combines spectral methods, group-theoretic arguments, and techniques from the theory of association schemes. As an application, we derive closed-form expressions for the Lovász $\vartheta$-function of generalized-Hamming graphs and their complements whenever either the graph or its complement is edge-transitive. Overall, the results demonstrate how spectral methods provide powerful tools for understanding the structure and symmetry of graphs, and they suggest several directions for further research.

The celebrated Bollobás-Eldridge-Catlin packing conjecture states that every $n$-vertex graph $G$ with minimum degree at least $\big(1-\frac{1}{\Delta+1}\big) n$ contains every $n$-vertex graph $H$ of maximum degree at most $\Delta$. Despite considerable attention, the conjecture remains widely open. We show that for bipartite $H$ this threshold can be greatly improved: there is an absolute constant $c>0$ such that every $n$-vertex graph $G$ with minimum degree at least $ \big(1-c\frac{\log\Delta}{\Delta}\big)n $ contains every $n$-vertex bipartite graph $H$ of maximum degree at most $\Delta$, provided $\Delta$ is not too large compared to $n$. Moreover, we prove that this logarithmic improvement is best possible up to the value of the constant.

An eigenvalue of the signless Laplacian $Q(G)$ is $Q$-main if its eigenspace is not orthogonal to the all-ones vector. We characterize graphs with exactly $\ell\ge3$ $Q$-main eigenvalues, one of which is zero. The case $\ell=3$ reduces to non-semiregular bipartite graphs satisfying a vertexwise signed degree-sum identity. For each integer $k\ge0$, we construct infinitely many pairwise nonisomorphic graphs of cyclomatic number $k$ and unbounded diameter, all with exactly three $Q$-main eigenvalues including zero. These families provide counterexamples to the stated classifications of trees, unicyclic graphs, and bicyclic graphs of Javarsineh and Fath-Tabar.

Shi-Catalan and affine Weyl arrangements are obtained in a similar way: they consist of a Weyl arrangement embedded into a higher dimensional space together with shifted copies. We consider the more general case when the Weyl arrangement is replaced by an arbitrary finite arrangement of hyperplanes. Such an arrangement is called an arrangement with special vertex. As our main result we prove that the number of parallel classes of lines in certain simplicial arrangements with special vertex is at most 12.

We use partially ordered sets (posets) to provide a canonical parameterization for the Zariski closure of the image of a semialgebraic set under a rational map whose coordinate functions are polynomials with nonnegative integral coefficients. The resulting poset parametrization of such a unirational variety allows us to translate several well-studied problems into combinatorics; e.g. reducing the problems to describing the poset associated to the variety. These problems include, the implicitization problem from algebraic geometry, the toric reparameterization problem, the computation of the linear span of the variety, and the problem of distinguishing two semialgebraic subsets of the same ambient space. The technique applies to instances of these problems in several fields, including algebraic geometry, algebraic combinatorics, statistics and applied algebra. We demonstrate the technique on examples from each field, including degenerate subvarieties of secant varieties, matroid flat varieties – which generalize toric varieties of edge polytopes, as well as varieties arising in multivariate data analysis and evolutionary biology.

Cameron and Erdős asked if the number of sets free of arithmetic progressions of length $k$ is $2^{r_k(n)(1+o(1))}$, where $r_k(n)$ is the maximum cardinality of a $k$-AP-free subset of $\{1, \dots, n\}$. Balogh, Liu and Sharifzadeh made significant progress on this question showing that it is $2^{O(r_k(n))}$ for an infinite sequence of $n$. We improve their result in two ways. On the one hand, we prove that, for $k\geq 5$, the number of $k$-AP-free sets in $[n]$ is $2^{r_k(n)(1+o(1))}$ for an infinite sequence of $n$, solving the question of Cameron and Erdős for infinitely many values. On the other hand, we also prove that for $k \geq 3$ and all $n$ the number of $k$-AP-free sets in $[n]$ is $2^{O(r_k(n))}$. These results are in fact special cases of a general framework that we develop to count families of sets excluding certain arithmetic patterns, which applies as long as the corresponding extremal threshold satisfies certain Behrend-type lower bounds. As further examples, we get analogous results for solution sets to almost all systems of linear equations as well as counting versions of the multidimensional Szemerédi theorem.

For a finite commutative ring $R$, let $\GR$ denote its zero-divisor graph and $\WGR$ its weakly zero-divisor graph, the latter containing the former as a spanning subgraph. We introduce the weak zero-divisor difference graph $\DR:=\WGR-\GR$ and develop a complete structural theory for finite reduced rings $R\cong\mathbb F_{q_1}\times\cdots\times\mathbb F_{q_t}$. We show $\DR$ sits strictly between $\GR$ and $\WGR$ in a three-stage refinement that also contains Badawi's annihilator graph, and prove that distinct support classes $X_A,X_B$ are completely joined in $\DR$ if and only if $A\cap B\ne\emptyset$ – an exact criterion underlying every result that follows. Consequently $\DR$ is edgeless for $t\le2$ but connected with diameter $2$ and girth $3$ for every $t\ge3$, independently of the field orders. We establish a complete perfectness dichotomy – $\DR$ is perfect exactly when $t\in\{3,4\}$, with an elementary combinatorial proof at $t=4$, and never perfect for $t\ge5$ – and determine its clique number exactly at $t=3$ and $t=4$; for general $t$ we give two incomparable lower bounds and two upper bounds, sharp at $t=3$ but not beyond, together with a compression argument showing an extremal family may always be taken shifted without this alone resolving the problem. A reconstruction theorem shows $\DR$ recovers the multiset of field orders intrinsically, so $\DR\cong\mathcal D(S)$ forces $R\cong S$. We further give closed forms, valid for every $t\ge3$, for the degree sequence and minimum degree, the domination number, and the independence and vertex cover numbers. Finally, we briefly indicate, via the valuation structure of finite chain rings, why the reduced-ring hypothesis cannot simply be dropped.

We prove that for every non-arithmetic set~$C$ and every arithmetic finite coloring of~$\mathbb{N}$, there is an infinite set $H \subseteq \mathbb{N}$ whose non-empty finite sums of distinct elements is monochromatic, and $C$ is not $H$-computable. We also study restrictions of Hindman's theorem to simple colorings.

Recently, Merino and Mütze (FOCS'23+SICOMP'24) presented an algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope ${\rm conv}(X)$, where $X\subseteq\{0,1\}^n$. The algorithm uses as a black box an algorithm for solving the classical linear optimization problem $\min\{w\cdot x\mid x\in X\}$ for some weight vector $w\in\mathbb{R}^n$. The resulting delay per visited vertex on the Hamilton path is only by a $\log n$ factor larger than the time to solve one instance of the optimization algorithm. In this paper, we make the Hamilton path algorithm simpler and faster. Namely, we obtain an amortized delay that is only by a constant factor larger than the running time of the optimization algorithm, thus removing the $\log n$ factor. As concrete results, this yields improved algorithms for generating bases and independent sets in a matroid, spanning trees, forests, matchings and maximum matchings in a graph, vertex covers, minimum vertex covers, independent sets and maximum independent sets in a bipartite graph, and antichains, maximum antichains and ideals in a poset. All of these listings correspond to Hamilton paths on the corresponding polytopes. Furthermore, we obtain an $\mathcal{O}(t_{\rm LP})$ amortized delay algorithm for the vertex enumeration problem on 0/1-polytopes $\{x\in\mathbb{R}^n\mid Ax\leq b\}$, where $A\in \mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$, and $t_{\rm LP}$ is the time needed to solve the linear program $\min\{w\cdot x\mid Ax\leq b\}$. This improves upon the $\mathcal{O}(t_{\rm LP} \log n)$ delay algorithm of Merino and Mütze, and the previous $\mathcal{O}(t_{\rm LP}\,n)$ delay algorithm of Bussieck and Lübbecke from 1998.

A latin square of order $n$ is an $n\times n$ array in which each of $n$ symbols occurs exactly once in each row and column. A transversal in such a square is a selection of $n$ entries that includes one representative of each row and column, and one of each symbol. For all even orders $n\ge 28$ except $n=30$, we construct a latin square of order $n$ in which every pair of transversals share at least one entry. We conjecture that in our squares there is no single entry that is common to all transversals. We prove this conjecture for $n\le10\,000$ by finding transversals using an algorithm that is likely to be of independent interest. We say that a transversal is dominant if it intersects every other transversal of the same latin square. We show that there exist latin squares of order $n$ that have a dominant transversal for $n\in\{5,7\}$ and also for all $n\ge8$ such that $n\not\equiv3\bmod4$.

In 1969, Halin proved that every $k$-connected graph $G$ with minimum degree at least $k+1$ contains an edge $e$ such that $G-e$ is $k$-connected. As an edge is a matching of size one, it is natural to ask whether Halin's result extends to matchings of larger size, a question recently investigated by Li, Zhou, Fujita, and Mao. A matching $M$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-M$ is $k$-connected. In this paper, we study minimum degree conditions that guarantee the existence of a $k$-removable matching of prescribed size. Specifically, we prove that for all positive integers $k$ and $m$, every $k$-connected graph $G$ with at least $2m$ vertices contains a $k$-removable matching of size $m$ if \[\delta(G)\ \ge\ \begin{cases} \max\bigl\{k+\bigl\lceil\tfrac m2\bigr\rceil,\ 2m\bigr\} & \text{if } k\ge m,\\[2pt] k+m & \text{if } k<m. \end{cases}\] As a consequence, every $k$-connected graph $G$ with $\delta(G)\ge2k+1$ contains a $k$-removable matching of size $\bigl\lceil(\delta(G)+1)/2\bigr\rceil$, unless $\delta(G)$ is even and $G\cong K_{\delta(G)+1}$. This verifies a conjecture of Li, Zhou, Fujita, and Mao in the range $\delta(G)\ge2k+1$. Our main tool, of independent interest, is a strengthening of Halin's result producing a $k$-removable edge that avoids a prescribed set of vertices.

This article studies the intersection of two linear sets of rank 3 of PG(1,q^3). In particular, all possible intersection sizes are determined, the structure of the largest intersection sets is described, and existence results are given.

The Kourovka Notebook is a long-running collection of open problems in group theory. In this paper we present solutions to eight of its problems. We construct a group with exactly two maximal locally soluble normal subgroups and show that, for every $1 \le k\le n!$, there is a group containing $n$ distinct elements whose $n!$ ordered products take exactly $k$ distinct values. We also give examples showing that group order together with the statistic $\sum_g\varphi(\lvert g\rvert)$ does not determine simplicity, and we construct a surjective non-injective Rota-Baxter operator on a non-abelian group. Further, we determine the group generated by the class transpositions of moduli at most $k$, prove that every power graph of a finite group that is a cograph is chordal, show that the right-relatively convex subgroups of a right-orderable group need not form a sublattice of its subgroup lattice, and disprove a proposed rank inequality for certain $p$-group extensions. All of these solutions were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.