Journal of Combinatorial Theory Series A最新文献

Let G be the alternating group of degree n. Let $\omega \left(G\right)$ be the maximal size of a subset S of G such that $〈x,y〉=G$ whenever $x,y\in S$ and $x\ne y$ and let $\sigma \left(G\right)$ be the minimal size of a family of proper subgroups of G whose union is G. We prove that, when n varies in the family of composite numbers, $\sigma \left(G\right)/\omega \left(G\right)$ tends to 1 as $n\to \infty$. Moreover, we explicitly calculate $\sigma \left(A_n\right)$ for $n\ge 21$ congruent to 3 modulo 18.

A plane curve $C\subset P^2$ of degree d is called blocking if every $F_q$-line in the plane meets C at some $F_q$-point. We prove that the proportion of blocking curves among those of degree d is $o\left(1\right)$ when $d\ge 2q-1$ and $q\to \infty$. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d\ge 3p$ and $d,q\to \infty$. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of $F_q$-roots of random polynomials, we find that the limiting distribution of the number of $F_q$-points in the intersection of a random plane curve and a fixed $F_q$-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of $F_q$-points contained in a union of k lines for $k=1,2,\dots ,q^2+q+1$.

A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph $K_n\setminus C_k$, obtained from the complete graph $K_n$ with n vertices by deleting all edges of a cycle $C_k$ with k vertices, is determined by its spectrum for $k\in \{3,4,5\}$, but not for $k=6$. In this paper, we show that $k=6$ is the unique exception for the spectral determination of $K_n\setminus C_k$.

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

It is well-known that for a prime $p\equiv 2\left(\mathrm{mod}3\right)$ and integer $n\ge 1$, the maximum possible size of a sum-free subset of the elementary abelian group $Z_p^n$ is $\frac{1}{3}(p+1)p^{n-1}$. However, the matching stability result is known for $p=2$ only. We consider the first open case $p=5$ showing that if $A\subseteq Z_5^n$ is a sum-free subset with $\left|A\right|>\frac{3}{2}\cdot 5^{n-1}$, then there are a subgroup $H<Z_5^n$ of size $\left|H\right|=5^{n-1}$ and an element $e\notin H$ such that $A\subseteq (e+H)\cup (-e+H)$.

More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.

Gyárfás famously showed that in every r-coloring of the edges of the complete graph $K_n$, there is a monochromatic connected component with at least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for k-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of $1+o\left(1\right)$ as the number of colors grows. We further generalize these questions to ask about counts of vertex s-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.

Consider a Borcherds-Kac-Moody Lie superalgebra, denoted as $g$, associated with the graph G. This Lie superalgebra is constructed from a free Lie superalgebra by introducing three sets of relations on its generators: (1) Chevalley relations, (2) Serre relations, and (3) The commutation relations derived from the graph G.

The Chevalley relations lead to a triangular decomposition of $g$ as $g=n_{+}\oplus h\oplus n_{-}$, where each root space $g_{\alpha }$ is contained in either $n_{+}$ or $n_{-}$. Importantly, each $g_{\alpha }$ is determined solely by relations (2) and (3). This paper focuses on the root spaces of $g$ that are unaffected by the Serre relations. We refer to these root spaces as “free roots” of $g$ (these root spaces are free from the Serre relations and can be associated with certain grade spaces of freely partially commutative Lie superalgebras, as detailed in Lemma 3.10. Consequently, we refer to them as “free roots,” and the corresponding root spaces in $g$ as “free root spaces” [cf. Definition 2.6]). Since these root spaces only involve commutation relations derived from the graph G, we can examine them purely from a combinatorial perspective.

We employ heaps of pieces to analyze these root spaces and establish various combinatorial properties. We develop two distinct bases for these root spaces of $g$: We extend Lalonde's Lyndon heap basis, originally designed for free partially commutative Lie algebras, to accommodate free partially commutative Lie superalgebras. We expand upon the basis introduced in the reference [1], designed for the free root spaces of Borcherds algebras, to encompass BKM superalgebras. This extension is achieved by investigating the combinatorial properties of super Lyndon heaps. Additionally, we also explore several other combinatorial properties related to free roots.

In this paper, we construct a bijection between 3142-avoiding permutations and 3241-avoiding permutations which proves the equidistribution of five classical set-valued statistics. Our bijection also enables us to establish a bijection between 3142-avoiding permutations and 4132-avoiding permutations, and a bijection between 2413-avoiding permutations and 1423-avoiding permutations, both of which preserve five classical set-valued statistics. Our results are generalizations of several conjectures posed by Burstein.