Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic analysis of $k$-noncrossing matchings
299
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set ${1,...,2n}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject to the condition that there exist no $k$ arcs that mutually intersect. We derive: (a) for arbitrary $k$, an asymptotic approximation of the exponential generating function of $k$-noncrossing matchings $F_k(z)$. (b) the asymptotic formula for the number of $k$-noncrossing matchings $f_{k}(n) sim c_k n^{-((k-1)^2+(k-1)/2)} (2(k-1))^{2n}$ for some $c_k>0$.
rate research
Read More
In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contains any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA pseudoknot structures cite{Stadler:99,Reidys:07pseu,Reidys:07lego} and their properties reflect basic features of these bio-molecules. The particular case of modular noncrossing diagrams has been extensively studied cite{Waterman:78b, Waterman:79,Waterman:93, Schuster:98}. Let ${sf Q}_k(n)$ denote the number of modular $k$-noncrossing diagrams over $n$ vertices. We derive exact enumeration results as well as the asymptotic formula ${sf Q}_k(n)sim c_k n^{-(k-1)^2-frac{k-1}{2}}gamma_{k}^{-n}$ for $k=3,..., 9$ and derive a new proof of the formula ${sf Q}_2(n)sim 1.4848, n^{-3/2},1.8489^{-n}$ cite{Schuster:98}.
A set partition is said to be $(k,d)$-noncrossing if it avoids the pattern $12... k12... d$. We find an explicit formula for the ordinary generating function of the number of $(k,d)$-noncrossing partitions of ${1,2,...,n}$ when $d=1,2$.
In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of $k$-noncrossing tangled-diagrams $T_{k}(n) sim c_k n^{-((k-1)^2+(k-1)/2)} (4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$.
A $k$-matching $M$ of a graph $G=(V,E)$ is a subset $Msubseteq E$ such that each connected component in the subgraph $F = (V,M)$ of $G$ is either a single-vertex graph or $k$-regular, i.e., each vertex has degree $k$. In this contribution, we are interested in $k$-matchings within the four standard graph products: the Cartesian, strong, direct and lexicographic product. As we shall see, the problem of finding non-empty $k$-matchings ($kgeq 3$) in graph products is NP-complete. Due to the general intractability of this problem, we focus on distinct polynomial-time constructions of $k$-matchings in a graph product $Gstar H$ that are based on $k_G$-matchings $M_G$ and $k_H$-matchings $M_H$ of its factors $G$ and $H$, respectively. In particular, we are interested in properties of the factors that have to be satisfied such that these constructions yield a maximum $k$-matching in the respective products. Such constructions are also called well-behaved and we provide several characterizations for this type of $k$-matchings. Our specific constructions of $k$-matchings in graph products satisfy the property of being weak-homomorphism preserving, i.e., constructed matched edges in the product are never projected to unmatched edges in the factors. This leads to the concept of weak-homomorphism preserving $k$-matchings. Although the specific $k$-matchings constructed here are not always maximum $k$-matchings of the products, they have always maximum size among all weak-homomorphism preserving $k$-matchings. Not all weak-homomorphism preserving $k$-matchings, however, can be constructed in our manner. We will, therefore, determine the size of maximum-sized elements among all weak-homomorphims preserving $k$-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.
Let $K_{n}^{r}$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A starting point for this study comes from a well-known result of Alon, Frankl, and Lovasz (1986). Our motivation is to find the smallest $n$ such that every $t$-coloring of $K_{n}^{r}$ contains an $s$-colored matching of size $k$. It has been conjectured that in every coloring of the edges of $K_n^r$ with 3 colors there is a 2-colored matching of size at least $k$ provided that $n geq kr + lfloor frac{k-1}{r+1} rfloor$. The smallest test case is when $r=3$ and $k=4$. We prove that in every 3-coloring of the edges of $K_{12}^3$ there is a 2-colored matching of size 4.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
