The stability number alpha(G) of a graph G is the cardinality of a maximum stable set in G, xi(G) denotes the size of core(G), where core(G) is the intersection of all maximum stable sets of G. In this paper we prove that for a graph G without isolated vertices, the following assertions are true: (i) if xi(G)< 2, then G is quasi-regularizable; (ii) if G is of order n and alpha(G) > (n+k-1)/2, for some k > 0, then xi(G) > k, and xi(G) > k+1, whenever n+k-1 is even. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that alpha(G) > n/2 implies xi(G) > 0. G is a Koenig-Egervary graph if n equals the sum of its stability number and the cardinality of a maximum matching. For Koenig-Egervary graphs, we prove that alpha(G) > n/2 holds if and only if xi(G) is greater than the size of the neighborhood of core(G). Moreover, for bipartite graphs without isolated vertices, alpha(G) > n/2 is equivalent to xi(G) > 1. We also show that Hall's marriage Theorem is valid for Koenig-Egervary graphs, and it is sufficient to check Hall's condition only for one specific stable set, namely, for core(G).
We introduce two frameworks in order to deal with fractal and multi-fractal analysis for subset sum problems where some embedding into the $1$-dimensional Euclidean space plays an important role. As one of these frameworks, the notion of the combinatorial $q$-fractal dimension for a subset sum function is introduced. Thereby, ``non-classical'' generalized dimensions for a family of subset~sum functions can be defined. These generalized dimensions include the box-counting dimension, the information dimension and the correlation dimension as well as the classical case. The combinatorial $q$-frac…
本文介绍了\emph{Serial and Unserial Methods}（SUM）。该算法是密切相关的第一部分的经典参考组合，\emph{Combinatorial algorithms for computers and calculators}，从阿尔伯特Nijenhuis和赫伯特Wilf。\emph{Serial Method}的建议是从一个特定类型的组合族在所有组合可能性列表中的位置获得它的输出。\emph{Unserial Method}是串行方法的逆步骤，从作为输入的组合族中获得序列号。\emph{serial number}是组合族在列表中的位置。
Flag matroids are a rich family of Coxeter matroids that can be characterized using pairs of matroids that form a quotient. We consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorial representations to explore characterizations of flag positroids. Given a uniform positroid, we give a purely combinatorial characterization of a family of positroids that form quotients with it. We state this in terms of their associated decorated permutations. In proving our characterization we also fully describe the circuits of this family.
A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the…
In this paper, we consider the problem of computing the entire sequence of the maximum degree of minors of a block-structured symbolic matrix (a generic partitioned polynomial matrix) $A = (A_{αβ} x_{αβ} t^{d_{αβ}})$, where $A_{αβ}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$, $x_{αβ}$ is an indeterminate, and $d_{αβ}$ is an integer for $α= 1,2,\dots, μ$ and $β= 1,2,\dots,ν$, and $t$ is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight bipartite matching problem. The main result of this paper is a combinatorial $O(μν\min\{μ, ν\}…
The Al-Salam-Chihara polynomials are an important family of orthogonal polynomials in one variable $x$ depending on 3 parameters $α$, $β$ and $q$. They are closely connected to a model from statistical mechanics called the partially asymmetric simple exclusion process (PASEP) and they can be obtained as a specialization of the Askey-Wilson polynomials. We give two different combinatorial formulas for the coefficients of the (transformed) Al-Salam-Chihara polynomials. Our formulas make manifest the fact that the coefficients are polynomials in $α$, $β$ and $q$ with positive coefficients.
Multitriangulations, and more generally subword complexes, yield a large family of simplicial complexes that are homeomorphic to spheres. Until now, all attempts to prove or disprove that they can be realized as convex polytopes faced major obstacles. In this article, we lay out the foundations of a framework -- built upon notions from algebraic combinatorics and discrete geometry -- that allows a deeper understanding of geometric realizations of subword complexes of Coxeter groups. Namely, we describe explicitly a family of chirotopes that encapsulate the necessary information to obtain geome…
The aim of this paper is to give some combinatorial relations linked polynomials generalizing those of Appell type to the partial r-Bell polynomials. We give an inverse relation, recurrence relations involving some family of polynomials and their exact expressions at rational values in terms of the partial r-Bell polynomials. We illustrate the obtained results by various comprehensive examples.
The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line $\mathrm{PG}(1,q^n)$. In this paper, we provide a large family of new maximum scattered linear sets over $\mathrm{PG}(1,q^n)$ for any even $n\geq 6$ and odd $q$. In particular, the relevant family contains …
In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$.
We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colo…
In the early 1990s, a family of combinatorial CW-complexes named permutoassociahedra was introduced by Kapranov, and it was realized by Reiner and Ziegler as a family of convex polytopes. The polytopes in this family are "hybrids" of permutohedra and associahedra. Since permutohedra and associahedra are simple, it is natural to search for a family of simple permutoassociahedra, which is still adequate for a topological proof of Mac Lane's coherence. This paper presents such a family.
Guo and Xu determined the maximum size of intersecting families over finite affine spaces and showed that any family reaches maximum size must be trivial. In this paper, we characterize non-trivial intersecting family with maximum size.
We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019) to a more general family, proving that such linear sets are maximum scattered when $q$ is odd and, apart from a special case, they are are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters $(6,6,q;5)$.
Simple greedy 2-approximation algorithm for the maximum genus of a graph
图G的最大亏格$γ_M(G)$是图G具有胞腔嵌入的可定向曲面的最大亏格。在组合上，它与G的邻边的不相交对的最大数目一致，这些边的去除导致G的连通生成子图。本文证明了在保持连通性的情况下任意删除G的邻边对至少会删除$γ_M(G)/2$对边。这使我们能够描述一个贪婪算法的最大图亏格;我们的算法返回一个整数k，使$γ_M(G)/2\le k \le γ_M(G)$，提供了一个简单的方法，有效地近似最大亏格。作为我们的方法的结果，我们得到了一个2-近似对应的Xuong的组合特征的最大亏格。
For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in \mathcal F$. Let $\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \cap \{1, 2, 3\}| \geq 2$. We prove that for $n \geq 50 k^2$ $|\mathcal I(\mathcal F)|$ is maximized by $\mathcal A$.
We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations listed in the paper by Bernstein & Sloane, now seen as transformations under the umbrella of partial Bell polynomials. Our goal is to describe these transformations from the algebraic and combinatorial points of view. We provide functional equations satisfied by the generating functions, derive inverse relations, and give a convolution formula. While the full …
Given an arbitrary hypergraph $\mathcal{H}$, we may glue to $\mathcal{H}$ a family of hypergraphs to get a new hypergraph $\mathcal{H}'$ having $\mathcal{H}$ as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph $\mathcal{H}'$ is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying ni…
Combinatorial transfer: a new method for constructing infinite families of nonabelian difference sets, partial difference sets, and relative difference sets
