Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn extremal problem on crossing vectors
285
0
0.0
(
0
)
Added by
Bartosz Walczak
Publication date
2012
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
For positive integers $w$ and $k$, two vectors $A$ and $B$ from $mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]geq k$ and $B[j]-A[j]geq k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in $mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $wleq 3$ and provide weaker upper bounds for $wgeq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.
rate research
Read More
A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a graph $G$. Such a question clearly needs restrictions on the colorings to be meaningful. For edge-colorings using $n-1$ colors and without rainbow cycles, known in the literature as JL-colorings, there turns out to be a particularly nice way of counting the rainbow spanning trees and we solve this problem completely for JL-colored complete graphs $K_n$ and complete bipartite graphs $K_{n,m}$. In both cases, we find tight upper and lower bounds; the lower bound for $K_n$, in particular, proves to have an unexpectedly chaotic and interesting behavior. We further investigate this question for JL-colorings of general graphs and prove several results including characterizing graphs which have JL-colorings achieving the lowest possible number of rainbow spanning trees. We establish other results for general $n-1$ colorings, including providing an analogue of Kirchoffs matrix tree theorem which yields a way of counting rainbow spanning trees in a general graph $G$.
An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index ($textrm{sz}(G)$). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved that the star is a tree having the maximal $textrm{sz}(G)$. Finding a tree with the minimal $textrm{sz}(G)$ is not an easy task to be done. Here, we present the minimal trees up to 25 vertices obtained by computer and describe the regularities which retain in them. Our preliminary computer tests suggest that a tree with the minimal $textrm{sz}(G)$ is also the connected graph of the given order that attains the minimal weighted Szeged index. Additionally, it is proven that among the bipartite connected graphs the complete balanced bipartite graph $K_{leftlfloor n/2rightrfloorleftlceil n/2 rightrceil}$ attains the maximal $textrm{sz}(G)$,. We believe that the $K_{leftlfloor n/2rightrfloorleftlceil n/2 rightrceil}$ is a connected graph of given order that attains the maximum $textrm{sz}(G)$.
The bloom of complex network study, in particular, with respect to scale-free ones, is considerably triggering the research of scale-free graph itself. Therefore, a great number of interesting results have been reported in the past, including bounds of diameter. In this paper, we focus mainly on a problem of how to analytically estimate the lower bound of diameter of scale-free graph, i.e., how small scale-free graph can be. Unlike some pre-existing methods for determining the lower bound of diameter, we make use of a constructive manner in which one candidate model $mathcal{G^*} (mathcal{V^*}, mathcal{E^*})$ with ultra-small diameter can be generated. In addition, with a rigorous proof, we certainly demonstrate that the diameter of graph $mathcal{G^{*}}(mathcal{V^{*}},mathcal{E^{*}})$ must be the smallest in comparison with that of any scale-free graph. This should be regarded as the tight lower bound.
Let $G$ be a 3-partite graph with $k$ vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matouu{s}ek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with $(1 - o(1)) k^{3/2} $ triangles, and a double counting argument shows that one cannot have more than $(1+o(1)) k^{7/4} $ triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to $(1 - o(1)) k^{5/3}$.
Wilfs Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as emph{Wilf partitions}. Using $f(n)$ to denote the number of Wilf partitions, we establish lead-order asymptotics for $ln{f(n)}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
