CyberSec Research

Generalizing works of D'Angeli and Donno, we describe, starting from an infinite sequence over $r$ letters with $r \neq 4i$ and $i \in \mathbb{N}$, a sequence of pointed finite graphs. We study the pointed Gromov-Hausdorff limit graphs giving a description of isomorphim classes in terms of dihedral groups and providing insights on the horofunction boundaries in terms of Busemann and non-Busemann points.

A perfect code in a graph $\Gamma=(V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we undertake a systematic study of which maximal subgroups of a group can be perfect codes. Our approach highlights a characterization of subgroup perfect codes in terms of their ``local'' complements.

Let $F_h(n)$ denote the minimum cardinality of an additive {\em $h$-fold basis} of $\{1,2,\cdots,n\}$: a set $S$ such that any integer in $\{1,2,\cdots, n\}$ can be written as a sum of at most $h$ elements from $S$. While the trivial bounds $h!n \; \lesssim \; F_h(n)^h \; \lesssim \; h^h n$ are well-known, comparatively little has been established for $h>2$. In this paper, we make significant improvements to both of the best-known bounds on $F_h(n)$ for sufficiently large $h$. For the lower bound, we use a probabilistic approach along with the Berry-Esseen Theorem to improve upon the best-known asymptotic result due to Yu. We also establish the first nontrivial asymptotic upper bound on $F_h(n)$ by leveraging a construction for additive bases of finite cyclic groups due to Jia and Shen. In particular, we show that given any $\epsilon>0$, for sufficiently large $h$, we have \[ \left(\frac{1}{2}-\epsilon\right)h!\sqrt{2\pi e} n\; \leq \; F_h(n)^h \; \leq \; \left(\left(\frac{\sqrt{3}}{2}+\epsilon\right)h\right)^h n. \]

In 1847, Kirkman proved that there exists a Steiner triple system on $n$ vertices (equivalently a triangle decomposition of the edges of $K_n$) whenever $n$ satisfies the necessary divisibility conditions (namely $n\equiv 1,3 \mod 6$). In 1970, Nash-Williams conjectured that every graph $G$ on $n$ vertices with minimum degree at least $3n/4$ (for $n$ large enough and satisfying the necessary divisibility conditions) has a triangle decomposition. In 1973, Erd\H{o}s conjectured that for each integer $g$, there exists a Steiner triple system on $n$ vertices with girth at least $g$ (provided that $n\equiv 1,3 \mod 6$ is large enough compared to the fixed $g$). In 2021, Glock, K\"uhn, and Osthus conjectured the common generalization of these two conjectures, dubbing it the ``Erd\H{o}s meets Nash-Williams' Conjecture''. In this paper, we reduce the combined conjecture to the fractional relaxation of the Nash-Williams' Conjecture. Combined with the best known fractional bound of Delcourt and Postle, this proves the combined conjecture above when $G$ has minimum degree at least $0.82733n$. We note that our result generalizes the seminal work of Barber, K\"uhn, Lo, and Osthus on Nash-Williams' Conjecture and the resolution of Erd\H{o}s' Conjecture by Kwan, Sah, Sawhney, and Simkin. Both previous proofs of those results used the method of iterative absorption. Our proof instead proceeds via the newly developed method of refined absorption (and hence provides new independent proofs of both results).

We construct a 3-uniform 1-degenerate hypergraph on $n$ vertices whose 2-colour Ramsey number is $\Omega\big(n^{3/2}/\log n\big)$. This shows that all remaining open cases of the hypergraph Burr-Erd\H{o}s conjecture are false. Our graph is a variant of the celebrated hedgehog graph. We additionally show near-sharp upper bounds, proving that all 3-uniform generalised hedgehogs have 2-colour Ramsey number $O\big(n^{3/2}\big)$.

We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in $1/\pi$ with rational coefficients; the proof is effective and provides an algorithm to explicitly compute these sums. Along the way we introduce parametric liftings of our sums, and show that they are polynomials in the complete elliptic integrals of the first and second kind. Moreover, the degrees of these polynomials are at most half of the number of vertices of the tree. The computation of these tree-indexed sums is motivated by the study of large meandric systems, which are non-crossing configurations of loops in the plane.

Linear resolutions and the stronger notion of linear quotients are important properties of monomial ideals. In this paper, we fully characterize linear resolutions and linear quotients in terms of the lcm-lattice of monomial ideals. These results complement characterizations of these two properties in terms of the Alexander dual of the corresponding Stanley-Reisner simplicial complex. In addition, we discuss applications to the case of edge ideals.

Let $G$ be a bridgeless graph. We introduce the maximum edge girth of $G$, denoted by $g^*(G)=\max\{l_G(e)\mid e\in E(G)\}$, where $l_G(e)$ is the edge girth of $e$, defined as the length of the shortest cycle containing $e$. Let $F(d,A)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ and $g^*(G)\in A$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $F(d,A)$. Let $f(d)=F(d,A)$, where $A=\{a\in \mathbb{N}\mid 2\leq a\leq 2d+1\}$. Chv\'atal and Thomassen (JCT-B, 1978) obtained general bounds for $f(d)$ and showed that $f(2)=6$. Kwok et al. (JCT-B, 2010) proved that $9\leq f(3)\leq 11$. Wang and Chen (JCT-B, 2022) determined $f(3)=9$. In this paper, we give that $12\leq F(4,A^*)\leq 13$, where $A^*=\{2,3,6,7,8,9\}$.

We study the Fuss--Catalan algebras, which are generalizations of the Temperley--Lieb algebra and act on generalized Dyck paths, through non-crossing partitions. First, the Temperley--Lieb algebra is defined on non-crossing partitions, and a bijection between a Dyck path and a non-crossing partition is shown to be compatible with the Temperley--Lieb algebra on Dyck paths, or equivalently chord diagrams. We show that the Kreweras endomorphism on non-crossing partitions is equivalent to the rotation of chord diagrams under the bijection. Secondly, by considering an increasing $r$-chain in the graded lattice of non-crossing partitions, we define the Fuss--Catalan algebras on increasing $r$-chains. Through a bijection between an increasing $r$-chain and a generalized Dyck path, one naturally obtains the Fuss--Catalan algebra on generalized Dyck paths. As generalizations of the Fuss--Catalan algebra, we introduce the one- and two-boundary Fuss--Catalan algebras. Increasing $r$-chains of symmetric non-crossing partitions give symmetric generalized Dyck paths by the bijection, and the boundary Fuss--Catalan algebras naturally act on them. We show that these representations are compatible with the diagrammatic representations of the algebras by use of generalized chord diagrams. Thirdly, we discuss the integrability of the Fuss--Catalan algebras. For the Fuss--Catalan algebras with boundaries, we obtain a new solution of the reflection equation in the case of $r=2$.

In this paper, we determine all connected net-regular strongly regular signed graphs with degree 5. There are five and two strongly regular signed graphs with net-degree 3 and 1, respectively.

Every maximum scattered linear set in $\mathrm{PG}(1,q^5)$ is the projection of an $\mathbb{F}_q$-subgeometry $\Sigma$ of $\mathrm{PG}(4,q^5)$ from a plane $\Gamma$ external to the secant variety to $\Sigma$. The pair $(\Gamma,\Sigma)$ will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in $\mathrm{PG}(1,q^5)$, namely those of pseudoregulus and LP type, have been characterized in the literature by B. Csajb\'{o}k, C. Zanella in 2016 and by C. Zanella, F. Zullo in 2020. Let $(\Gamma,\Sigma)$ be a projecting configuration for a maximum scattered linear set in $\mathrm{PG}(1,q^5)$, let $\sigma$ be a generator of $\mathbb{G}=\mathrm{P}\Gamma \mathrm{L}(5,q^5)_\Sigma$, and $A=\Gamma\cap\Gamma^{\sigma^4}$, $B=\Gamma\cap\Gamma^{\sigma^3}$. If $A$ and $B$ are not both points, then the projected linear set is of pseudoregulus type. Then, suppose that they are points. The rank of a point $X$ is the vectorial dimension of the span of the orbit of $X$ under the action of $\mathbb{G}$. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points $A$ and $B$ has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that $\mathrm{rk} A=\mathrm{rk} B=4$. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for $q\leq 25$, no new maximum scattered linear set exists.

Resolvable combinatorial designs including Resolvable Balanced Incomplete Block Designs, Resolvable Group Divisible Designs, Uniformly Resolvable Designs and Mutually Orthogonal Latin Squares and Rectangles are used to construct optimal solutions to the Social Golfer problem (SGP) and the Social Golfer problem with adjacent group sizes (SGA). An algorithm is presented to find an optimal solution in general, and a complete set of solutions is provided for up to 150 players.

The arrow relation, a central concept in extremal set theory, captures quantitative relationships between families of sets and their traces. Formally, the arrow relation $(n, m) \rightarrow (a, b)$ signifies that for any family $\mathcal{F} \subseteq 2^{[n]}$ with $|\mathcal{F}| \geqslant m$, there exists an $a$-element subset $T \subseteq [n]$ such that the trace $\mathcal{F}_{|T} = \{ F \cap T : F \in \mathcal{F} \}$ contains at least $b$ distinct sets. This survey highlights recent progress on a variety of problems and results connected to arrow relations. We explore diverse topics, broadly categorized by different extremal perspectives on these relations, offering a cohesive overview of the field.

Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this paper, we study their cohomology by relating them to the cohomology of Hessenberg varieties in the full flag varieties.

We consider critical multitype Bienaym\'e trees that are either irreducible or possess a critical irreducible component with attached subcritical components. These trees are studied under two distinct conditioning frameworks: first, conditioning on the value of a linear combination of the numbers of vertices of given types; and second, conditioning on the precise number of vertices belonging to a selected subset of types. We prove that, under a finite exponential moment condition, the scaling limit as the tree size tends to infinity is given by the Brownian Continuum Random Tree. Additionally, we establish strong non-asymptotic tail bounds for the height of such trees. Our main tools include a flattening operation applied to multitype trees and sharp estimates regarding the structure of monotype trees with a given sequence of degrees.

This paper introduces two canonical constructions that transform arbitrary finite graphs into perfect graphs: the symmetric lift $\mathrm{HL}'_2(G)$, which is purely structural and label-invariant, and the ordered lift $\mathrm{HL}_2(G)$, which depends explicitly on vertex labeling and encodes directional information. Both lifts arise as line graphs of bipartite double covers and are box-perfect. The symmetric lift $\mathrm{HL}'_2(G)$ forms a canonical 2-cover of the line graph $L(G)$. This involution decomposes $\mathrm{HL}'_2(G)$ into symmetric and antisymmetric components: the symmetric part recovers $L(G)$, while the antisymmetric part yields a signed graph $L^-(G)$, the antisymmetric line graph, with +1/-1 edges encoding consistent vs. crossed overlaps. Thus, all adjacency and Laplacian eigenvalues of $L(G)$, with multiplicities, appear within those of $\mathrm{HL}'_2(G)$, despite $L(G)$ typically not being a subgraph. For regular graphs such as Paley graphs, this yields infinite families of sparse, highly structured regular and box-perfect expanders that also retain large cliques. The lift retains much of the spectral expansion of the base while improving the combinatorial expansion. Much the same behavior is observed with random regular base graphs, allowing for the possibility of the study of box-perfect random regular graphs. Finally, we generalize these constructions to parameterized lifts $\mathrm{HL}_{r,d}(G)$ and $\mathrm{HL}_{r,d}'(G)$ defined on ordered $r$-tuples connected by Hamming distance constraints, which structurally encode the base graph and remain box-perfect.

We introduce the new concept of weighted $K$-$k$-Schur functions -- a novel family within the broader class of Katalan functions -- that unifies and extends both $K$-$k$-Schur functions and closed $k$-Schur Katalan functions. This new notion exhibits a fundamental alternating property under certain conditions on the indexed $k$-bounded partitions. As a central application, we resolve the $K$-$k$-Schur alternating conjecture -- posed by Blasiak, Morse, and Seelinger in 2022 -- for a wide class of $k$-bounded partitions, including all strictly decreasing $k$-bounded partitions. Our results shed new light on the combinatorial structure of $K$-theoretic symmetric functions.

We prove that for every bipartite graph $H$ and positive integer $s$, the class of $K_{s,s}$-subgraph-free graphs excluding $H$ as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every $K_{s,t}$-free bipartite circle graph with $s\le t$ has a vertex of degree at most $\max\{2s-2, t-1\}$ and provide examples showing that this is tight.