On minimal complements in groups

On additive co-minimal pairs

A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is an additive complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We stu…

Asymptotic behaviour of minimal complements

The notion of minimal complements was introduced by Nathanson in 2011 as a natural group-theoretic analogue of the metric concept of nets. Given two non-empty subsets $W,W'$ in a group $G$, the set $W'$ is said to be a complement to $W$ if $W\cdot W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The inverse problem asks which sets may or not occur as minimal complements. We show some new results on the inverse problem and investigate how the study of the inverse problem naturally gives rise to questions about the asymptotic behaviour of these sets, providing partial …

Asymptotic complements in the integers

Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be a minimal asymptotic complement if $C$ is an asymptotic complement, but $C\setminus \lbrace c\rbrace$ is not an asymptotic complement $\forall c\in C$. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathan…

互补于最小素图的5环

5-cycles in the complement of minimal prime graphs

本文研究了最小素图补图的循环结构，证明了补图中每条边都包含在5环中，强化了关于诱导5环的存在性结论。

On Eventually Periodic Sets as Minimal Additive Complements

We say a subset $C$ of an abelian group $G$ \textit{arises as a minimal additive complement} if there is some other subset $W$ of $G$ such that $C+W=\{c+w:c\in C,\ w\in W\}=G$ and such that there is no proper subset $C'\supset C$ such that $C'+W=G$. In their recent paper, Burcroff and Luntzlara studied, among many other things, the conditions under which "eventually periodic sets", which are finite unions of infinite (in the positive direction) arithmetic progressions and singletons, arise as minimal additive complements in $\mathbb Z$. In the present paper we shall study this question further…

二度顶点在极小素图补图中的作用

Degree 2 vertices in minimal prime graph complements

本文研究极小素图补图中二度顶点的结构，证明其结构为有限重复的5-cycle，属于reseminant图类。

On dominating graph of graphs, median graphs and partial cubes, and graphs in which complement of every minimal dominating set is minimal dominating

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating graph has become an increasingly interesting subject in domination theory. On the other hand, median graphs and partial cubes are two fundamental graph classes in graph theory. In this paper, we make some new connections between domination theory and the theory of median graphs and partial cubes. As the main result, we show that the following conditions are eq…

The Boolean intervals of Chevalley type are strongly non group-complemented

Let G be a finite Chevalley group and B a Borel subgroup. Then the interval [B,G] in L(G) is Boolean. We prove, using Zsigmondy's theorem, that for any element P in the open interval (B,G), its lattice-complement P^c is not a group-complement.

The Smith group and the critical group of the Grassmann graph of lines in finite projective space and of its complement

We compute the elementary divisors of the adjacency and Laplacian matrices of the Grassmann graph on $2$-dimensional subspaces in a finite vector space. We also compute the corresponding invariants of the complementary graphs.

A differential algebra and the homotopy type of the complement of a toric arrangement

We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.

关于有限群幂图补图直径的注记

Notes on the diameter of the complement of the power graph of a finite group

该研究确定了有限群幂图补图与增强幂图补图各连通分支的直径，完整解答了Peter J. Cameron提出的两个相关问题。

On minimal degree of transitive permutation groups with stabiliser being a $2$-group

The minimal degree of a permutation group $G$ is defined as the minimal number of non-fixed points of a non-trivial element of $G$. In this paper we show that if $G$ is a transitive permutation group of degree $n$ having no non-trivial normal $2$-subgroups such that the stabiliser of a point is a $2$-group, then the minimal degree of $G$ is at least $\frac{2}{3}n$. The proof depends on the classification of finite simple groups.

On the fundamental group of the complement of a complex hyperplane arrangement

We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the complement to a line arrangement of a given combinatorial type with respect to isomorphisms inducing the canonical isomorphism of the first homology groups.

Euclidean matchings and minimality of hyperplane arrangements

We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us t…