László A. Székely

University of South Carolina | USC · Department of Mathematics

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide formulae for the minimum Wiener index of simple 5-connected triangulations and 3-connected quadrangulations, and provide the extremal structures, which attain those values. Our main tool is setting upper bounds for the maximum degre...

Erdős et al. made conjectures for the maximum diameter of connected graphs without a complete subgraph Kk + 1 ${K}_{k+1}$, which have order n $n$ and minimum degree δ $\delta $. Settling a weaker version of a problem, by strengthening the Kk + 1 ${K}_{k+1}$‐free condition to k $k$‐colorable, we solve the problem for k = 3 $k=3$ and k = 4 $k=4$ usin...

P. Erd\H{o}s, J. Pach, R. Pollack, and Z. Tuza [J. Combin. Theory, B 47 (1989), 279--285] made conjectures for the maximum diameter of connected graphs without a complete subgraph $K_{k+1}$, which have order $n$ and minimum degree $\delta$. Settling a weaker version of a problem, by strengthening the $K_{k+1}$-free condition to $k$-colorable, we so...

It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745–756] that for n≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 5$$\end{document} every simple...

A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing,...

Let $G$ be a connected graph. The {\sl Wiener index} of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide formulae for the minimum Wiener index of simple triangulations and quadrangulations with given connectedness, and provide the extremal structures which attain these values. As a main tool, we prov...

Let $ G $ be a connected graph. If $\bar{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the proximity, $\pi(G)$, of $G$ is defined as the smallest value of $\bar{\sigma}(v)$ over all vertices $v$ of $G$. We give upper bounds for the proximity of simple triangulations and quadrangulations of give...

We find the asymptotic behavior of the Steiner k-diameter of the $n$-cube if $k$ is large. Our main contribution is the lower bound, which utilizes the probabilistic method.

The crossing number CR ( G ) of a graph G = ( V , E ) is the smallest number of edge crossings over all drawings of G in the plane. For any k ≥ 1, the k‐planar crossing number of G , CR k ( G ), is defined as the minimum of CR ( G 1 ) + CR ( G 2 ) + ⋯ + CR ( G k ) over all graphs G 1 , G 2 , … , G k with ∪ i = 1 k G i = G. Pach et al [Comput. Geom....

Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and...

For n∈N let δn be the smallest value such that every graph of order n and minimum degree at least δn admits an orientation of diameter two. We show that δn=[Formula presented]+Θ(lnn).

For positive integers $n$ and $e$, let $\kappa(n,e)$ be the minimum crossing number (the standard planar crossing number) taken over all graphs with $n$ vertices and at least $e$ edges. Pach, Spencer and T\'oth [Discrete and Computational Geometry 24 623--644, (2000)] showed that $\kappa(n,e) n^2/e^3$ tends to a positive constant (called midrange c...

It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745--756] that for $n\geq 5$ every simple graph of order $n$ and size at least $\binom{n}{2}-n+5$ has an orientation of diameter two. We prove this conjecture and hence determine for every $n\geq 5$ the minimum value of $m$ such that every graph of order $n$ and size $m$ has an orienta...

The crossing number ${\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, ${\mbox {cr}}_k(G)$, is defined as the minimum of ${\mbox {cr}}(G_1)+{\mbox {cr}}(G_2)+\ldots+{\mbox {cr}}(G_{k})$ over all graphs $G_1, G_2,\ldots, G_{k}...

For $n \in \mathbb{N}$ let $\delta_n$ be the smallest value such that every graph of order $n$ and minimum degree at least $\delta_n$ admits an orientation of diameter two. We show that $\delta_n=\frac{n}{2} + \Theta(\ln n)$.

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of $d$-ary trees (rooted trees whose vertex outdegrees are bounded from above by $d\geq 2$) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For $T$ in the family of strictly $d$-ary trees (every vertex...

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges c...

Consider a set $W,$ natural numbers $d(w)$ associated with each $w\in W$, and a $W_i:i\in I$ partition of $W$ with natural numbers $c(W_i,W_j)$ associated to every unordered pair of partition classes. The Partition Adjacency Matrix realization problem asks whether there is a simple graph on the vertex set $W$ with degree sequence $d(w)$ for $w\in W...

A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing,...

Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector as a function of $n$. This provides an upper bound on the number of estimable parameters in the exponential ra...

We determine the maximum distance between any two of the center, centroid, and subtree core among trees with a given order. Corresponding results are obtained for trees with given maximum degree and also for trees with given diameter. The problem of the maximum distance between the centroid and the subtree core among trees with given order and diam...

The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown th...

In this chapter, we explore the history and the status of the Zarankiewicz crossing number conjecture and the Hill crossing number conjecture, on drawing complete bipartite and complete graphs in the plane with a minimum number of edge crossings. We discuss analogous problems on other surfaces and in different models of drawing.

In this chapter we explore recent development on various problems related to graph indices in trees. We focus on indices based on distances between vertices, vertex degrees, or on counting vertex or edge subsets of different kinds. Some of the indices arise naturally in applications, e.g., in chemistry, statistical physics, bioinformatics, and othe...

In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree $B$ with $k$ leaves, we let $\gamma(B,T)$ be the proportion of all subsets of $k$ leaves in $T$ that induce a tree isomorphic to $B$. The inducibility of $B$ is $\limsup_{|T| \to \infty} \gamma(B,T)$. We determine t...

We show that a tree of order $n$ has at most $O(5^{n/4})$ nonisomorphic subtrees, and that this bound is best possible. We also prove an analogous result for the number of nonisomorphic rooted subtrees of a rooted tree.

The aim of this paper is to provide an affirmative answer to a recent question by Bubeck and Linial on the local profile of trees. For a tree $T$, let $p^{(k)}_1(T)$ be the proportion of paths among all $k$-vertex subtrees (induced connected subgraphs) of $T$, and let $p^{(k)}_2(T)$ be the proportion of stars. Our main theorem states: if $p^{(k)}_1...

Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We study the maximum number of nonzero elements in a joint degree vector of an $n$-vertex graph. This provides an upper bound on the number of estimable parameters on a bidegree-distribution based exponential...

We study ensemble-based graph-theoretical methods aiming to approximate the size of the minimum dominating set (MDS) in scale-free networks. We analyze both analytical upper bounds of dominating sets and numerical realizations for applications. We propose two novel probabilistic dominating set selection strategies that are applicable to heterogeneo...

The eccentricity of a vertex, $ecc_T(v) = \max_{u\in T} d_T(v,u)$, was one of the first, distance-based, tree invariants studied. The total eccentricity of a tree, $Ecc(T)$, is the sum of eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios $Ecc(T)/ecc_T(u)$, $Ecc(T)/ecc_T(v)$, $ecc_T...

We discovered a dual behavior of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems (Székely and Wang, 2006, 2005). We introduced the concept of subtree core: the subtree core of a tree consists of one or two adjacent vertices of a tree that are contained in the largest number of subtrees. Let σ(T)σ(T)...

There is much recent interest in excluded subposets. Given a fixed poset $P$, how many subsets of $[n]$ can found without a copy of $P$ realized by the subset relation? The hardest and most intensely investigated problem of this kind is when $P$ is a {\sl diamond}, i.e. the power set of a 2 element set. In this paper, we show infinitely many asympt...

Our previous paper [the first and the third author, Electron. J. Comb. 14, No. 1, Research Paper R63, 13 p. (2007; Zbl 1183.05088)] applied a lopsided version of the Lovász local lemma that allows negative dependency graphs to the space of random matchings in K 2n , deriving new proofs to a number of results on the enumeration of regular graphs wit...

Aydinian et al. [J. Combinatorial Theory A 118(2)(2011), 702–725] substituted the usual BLYM inequality for L-Sperner families with a set of M inequalities for (m 1,m 2,…,m M ; L 1,L 2,…,L M ) type M-part Sperner families and showed that if all inequalities hold with equality, then the family is homogeneous. Aydinian et al. [Australasian J. Comb. 4...

The Lovász local lemma is a well-known probabilistic technique commonly used to prove the existence of rare combinatorial objects. We explore the lopsided (or negative dependency graph) version of the lemma, which, while more general, appears infrequently in literature due to the lack of settings in which the additional generality has thus far been...

We study four problems: put $n$ distinguishable/non-distinguishable balls into $k$ non-empty distinguishable/non-distinguishable boxes randomly. What is the threshold function $k=k(n) $ to make almost sure that no two boxes contain the same number of balls? The non-distinguishable ball problems are very close to the Erd\H os--Lehner asymptotic form...

In 1996, Guigo et al. [Mol. Phylogenet. Evol., 6 (1996), 189-203] posed the following problem: for a given species tree and a number of gene trees, what is the minimum number of duplication episodes, where several genes could have undergone duplication together to generate the observed situation. (Gene order is neglected, but duplication of genes c...

Aydinian et al. [J. Combinatorial Theory A 118(2)(2011), 702-725] substituted the usual BLYM inequality for L-Sperner families with a set of M inequalities for $(m_1,m_2,...,m_M;L_1,L_2,...,L_M)$ type M-part Sperner families and showed that if all inequalities hold with equality, then the family is homogeneous. Aydinian et al. [Australasian J. Comb...

Various topological indices have been put forward in different studies from bio-chemistry to pure mathematics. Among them the Wiener index, the number of subtrees and the Randić index have received great attention from mathematicians. While studying the extremal problems regarding these indices among trees, one interesting phenomenon is that they s...

In this paper we investigate common generalizations of more-part and L-Sperner families. We prove a BLYM inequality for M-part L-Sperner families and obtain results regarding the homogeneity of such families of maximum size through the convex hull method. We characterize those M-part Sperner problems, where the maximum family size is the classical...

P. L. Erdős and L. A. Székely [Adv. Appl. Math. 10, No. 4, 488–496 (1989; Zbl 0723.05046)] gave a bijection between rooted semilabeled trees and set partitions, which specializes to a bijection between phylogenetic trees and set partitions with classes of size ≥2. L. H. Harper’s results [Ann. Math. Stat. 38, 410–414 (1967; Zbl 0154.43703)] on the a...

We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs. We find a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We find the exact biplanar crossing number...

We investigate a packing problem in M-dimensional grids, where bounds are given for the number of allowed entries in different axis-parallel directions. The concept is motivated from error correcting codes and from more-part Sperner theory. It is also closely related to orthogonal arrays. We prove that some packing always reaches the natural upper...

There are three general lower bound techniques for the crossing numbers of graphs: the Crossing Lemma, the bisection method and the embedding method. In this contribution, we present their adaptations to the minor crossing number. Using the adapted bounds, we improve on the known bounds on the minor crossing number of hypercubes. We also point out...

This paper continues our earlier investigations into the inversion of random functions in a general (abstract) setting. In Section 2, we investigate a concept of invertibility and the invertibility of the composition of random functions defined on finite sets. In Section 3, we resolve some questions concerning the number of samples required to ensu...

We improve the lower bound on the extremal version of the Maximum Agreement Subtree problem. Namely we prove that two binary trees on the same n leaves have subtrees with the same ≥ c log log n leaves which are homeomorphic, such that homeomorphism is identity on the leaves.

Degree-based graph construction is a ubiquitous problem in network modelling (Newman et al 2006 The Structure and Dynamics of Networks (Princeton Studies in Complexity) (Princeton, NJ: Princeton University Press), Boccaletti et al 2006 Phys. Rep. 424 175), ranging from social sciences to chemical compounds and biochemical reaction networks in the c...

Let p be a graph parameter that assigns a positive integer value to every graph. The inverse problem for p asks for a graph within a prescribed class (here, we will only be concerned with trees), given the value of p. In this context, it is of interest to know whether such a graph can be found for all or at least almost all integer values of p. We...

We prove that for every connected 4-colourable graph G of order n and minimum degree � ≥ 1, diam(G) ≤ 5n 2� − 1. This is a first step toward proving a conjecture of Erdýos, Pach, Pollack and Tuza (4) from 1989.

Phylogenetic trees describe the evolutionary history of a group of present-day species from a common ancestor. These trees are typically reconstructed from aligned DNA sequence data. In this paper we analytically address the following question: Is the amount of sequence data required to accurately reconstruct a tree significantly more than the amou...

Our previous paper applied a lopsided version of the Lov\'asz Local Lemma that allows negative dependency graphs to the space of random injections from an $m$-element set to an $n$-element set. Equivalently, the same story can be told about the space of random matchings in $K_{n,m}$. Now we show how the cited version of the Lov\'asz Local Lemma app...

We give a pair of well-matched lower and upper bounds for the expectation of reversal distance under the hypothesis of random gene order by investigating the expected number of cycles in the breakpoint graph of linear signed permutations. Sankoff and Haque [9] proved similar results for circular signed permutations based on approximations based on...

The biplanar crossing number cr2(G) of a graph G is min{cr(G1) + cr(G2)}, where cr is the planar crossing number. We show that cr2(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) - 2 ≤ Kcr2(G)0.4057 log2n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time rand...

A sufficient condition is given that a certain drawing minimizes the cross- ing number. The condition is in terms of intersections in an arbitrary set system related to the drawing, and is like a correlation inequality.

The existence of Armstrong-instances of bounded domains is investigated for specific key systems. This leads to the concept of Armstrong(q,k,n)-codes. These are q-ary codes of length n, minimum distance n − k + 1 and have the property that for any possible k − 1 coordinate positions there are two codewords that agree exactly there. We derive upper...

A widely-studied model for generating binary sequences is to'evolve'them on a tree according to a symmetric Markov process. We show that under this model distinguishing the true (model) tree from a false one is substantially "easier" (in terms of the sequence length needed) than determining the true tree. The key tool is a new and tight Ramsey-type...

The Lov asz Local Lemma is known to have an extension for cases where inde- pendence is missing but negative dependencies are under control. We show that this is often the case for random injections, and we provide easy-to-check conditions for the non-trivial task of verifying a negative dependency graph for random injec- tions. As an application,...

Database motivations lead to the concept of SPT(q, k, n)-codes. These are q-ary codes of length n, minimum distance n−k+1 and have the property that for any possible k−1 coordinate positions there are two codewords that agree exactly there. We derive upper and lower bounds on the length of the code as function of q and k. The upper bounds use geome...

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K2k+1,q, for k⩾2. We prove tight bounds...

There are three general lower bound techniques for the crossing numbers of graphs, all of which can be traced back to Leighton's work on applications of crossing number in VLSI: the Crossing Lemma, the Bisection Method, and the Embedding Method. In this contribution, we sketch their adaptations to the minor crossing number.

This paper characterizes binary trees with n leaves, which have the greatest number of subtrees. These binary trees coincide with those which were shown by Fischermann et al. [Wiener index versus maximum degree in trees, Discrete Appl. Math. 122(1–3) (2002) 127–137] and Jelen and Triesch [Superdominance order and distance of trees with bounded maxi...

A widely studied model for generating sequences is to ``evolve'' them on a tree according to a symmetric Markov process. We prove that model trees tend to be maximally ``far apart'' in terms of variational distance.

The construction of evolutionary trees is a fundamental problem in biology, and yet methods for reconstructing evolutionary trees are not reliable when it comes to inferring accurate topologies of large divergent evolutionary trees from realistic length sequences. We address this problem and present a new polynomial time algorithm for reconstructin...

We study the bipartite crossing number problem. When the minimum degree and the maximum degree of the graph are close to each other, we derive two polynomial time approximation algorithms for solving this problem, with approximation factors, O(log2 n), and O(log n log log n), from the optimal, respectively, where n is the number of vertices. This p...

We prove that the crossing number of the cartesian product of 2 cycles, C m Cn, mn, is of order (mn), improving the best known lower bound. In particular we show that the crossing number of C mCn is at least mn/90, and for n=m, m+1 we reduce the constant 90 to 6. This partially answers a 20-years old question of Harary, Kainen and Schwenk [3] who g...

The paper introduces the book crossing number problem which can be viewed as a variant of the well-known plane and surface crossing number problem or as a generalization of the book embedding problem. The book crossing number of a graph G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page b...

The function lattice, or generalized Boolean algebra, is the set of ℓ-tuples with the ith coordinate an integer between 0 and a bound n i . Two ℓ-tuples t-intersect if they have at least t common nonzero coordinates. We prove a Hilton–Milner type theorem for systems of t-intersecting ℓ-tuples.

We study that over a certain type of trees (e.g., all trees or all binary trees) with a given number of vertices, which trees minimize or maximize the total number of subtrees (or subtrees with at least one leaf). Trees minimizing the total number of subtrees (or subtrees with at least one leaf) usually maximize the Wiener index, and vice versa. In...

Crossing numbers have drawn much attention in the last couple of years and several surveys [22], [28], [33], problem collections [26], [27], and bibliographies [40] have been published. The present survey tries to give pointers to some of the most significant recent developments and identifies computational challenges.

We extend the lower bound in [15] for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhib...

Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph K a1,a2, with b 1 roots in the first vertex class and b 2 roots in the second vertex clas s. We give a simple proof to their formula, and a generalization for complete m-partite graphs, using the multivariate Lagrange inverse.

The thickness of a graph G, is the minimum number of planar graphs, whose union is G. Halton conjectured that any graph of maximum degree d has thickness at most ⌈(d+2)/4⌉. We disprove the conjecture by showing graphs of thickness ⌈d/2⌉, for any d⩾5.

This paper surveys how the concept of crossing number, which used to be familiar only to a limited group of specialists, emerges as a significant graph parameter. This paper has dual purposes: first, it reviews foundational, historical, and philosophical issues of crossing numbers, second, it shows a new lower bound for crossing numbers. This new l...

A phylogenetic tree, also called an "evolutionary tree," is a leaf-labeled tree which represents the evolutionary history for a set of species, and the construction of such trees is a fundamental problem in biology. Here we address the issue of how many sequence sites are required in order to recover the tree with high probability when the sites ev...

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K 2 k + 1, q, for k ≥ 2. We prove tight...

A convex drawing of an n-vertex graph G = (V,E) is a drawing in which the vertices are placed on the corners of a convex n-gon in the plane and each edge is drawn using one straight line segment. We derive a general lower bound on the number of crossings in any convex drawings of G, using isoperimetric properties of G. The result implies that conve...

No abstract.

The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper we characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees of given order that have only vertices of two dierent degrees.

Adamec and Nešetřil [1] proposed a new the so called fractional length criterion for measuring the aesthetics of (artistic) drawings. They proposed to apply the criterion to the aesthetic drawing of graphs. In the graph drawing community, it is widely believed and even experimentally confirmed that the number of crossings is one of the most importa...

We study inverting random functions under the maximum likelihood estimation (MLE) criterion in the discrete setting. In particular, we consider how many independent evaluations of the random function at a particular element of the domain are needed for reliable reconstruction of that element. We provide explicit upper and lower bounds for MLE, both...

We provide counterexamples to two conjectures known in the field of graph drawing. The first conjecture (made by J. Halton ten years ago) asserts that the thickness of any graph of maximum degree Δ is at most ⌈(Δ+2)/4⌉. We give an existence proof that there are graphs of the thickness ⌈Δ/2⌉—this is known to be the best possible upper bound. The sec...

A computational method was developed for delineating connected gene neighborhoods in bacterial and archaeal genomes. These gene neighborhoods are not typically present, in their entirety, in any single genome, but are held together by overlapping, partially conserved gene arrays. The procedure was applied to comparing the orders of orthologous gene...

This paper actually did not give a citation to my work, although clearly ought to have done.) 5. H. Furstenberg, Y. Katznelson, B. Weiss, Ergodic theory and con- gurations in sets of positive density, in: Mathematics of Ramsey Theory, J. Nesetril, V. Rodl, eds., Algorithms and Combinatorics 5, Springer Verlag, 1990, 184-198

We study the integral uniform (multicommodity) flow problem in a graph G and construct a fractional solution whose properties are invariant under the action of a group of automorphisms Γ<Aut(G). The fractional solution is shown to be close to an integral solution (depending on properties of Γ), and in particular becomes an integral solution for a c...

Let G be a connected bipartite graph. We give a short proof, using a variation of Menger's Theorem, for a new lower bound which relates the bipartite crossing number of G, denoted by bcr(G), to the edge connectivity properties of G. The general lower bound implies a weaker version of a very recent result, establishing a bisection-based lower bound...

The bipartite crossing number problem is studied, and a connection between this problem and the linear arrangement problem is established. It is shown that when the arboricity is close to the minimum degree and the graph is not too sparse, then the optimal number of crossings has the same order of magnitude as the optimal arrangement value times th...

We prove Erdos-Ko-Rado and Hilton-Milner type theorems for t-intersecting k-chains in posets using the kernel method. These results are common generalizations of the original EKR and HM theorems, and our earlier results for intersecting k-chains in the Boolean algebra. For intersecting k-chains in the c-truncated Boolean algebra we also prove an ex...

We survey conjectured and proven Ahlswede-type higher-order generalizations of the Erdös-Ko-Rado theorem. This paper is dedicated to the 60th birthday of Professor Rudolf Ahlswede.

Inferring evolutionary trees is an interesting and important problem in biology, but one that is computationally difficult as most associated optimization problems are NP-hard. Although many methods are provably statistically consistent (i.e. the probability of recovering the correct tree converges to 1 as the sequence length increases), the actual...

We study the possibility of the existence of a Katona type proof for the Erdos-Ko-Rado theorem for 2- and 3-intersecting families of sets. An Erdos-Ko-Rado type theorem for 2-intersecting integer arithmetic progressions and a model theoretic argument show that such an approach works in the 2-intersecting case.

We study the possibility of the existence of a Katona type proof for the Erdos-Ko-Rado theorem for 2- and 3-intersecting families of sets. An Erdos-Ko-Rado type theorem for 2-intersecting integer arithmetic progressions and a model theoretic argument show that such an approach works in the 2-intersecting case. 1 Introduction One of the basic result...

A phylogenetic tree, also called an “evolutionary tree,” is a leaf-labeled tree which represents the evolutionary history for a set of species, and the construction of such trees is a fundamental problem in biology. Here we address the issue of how many sequence sites are required in order to recover the tree with high probability when the sites ev...

In this paper we study how to invert random functions under different criteria. The motivation for this study is phylogeny reconstruction, since the evolution of biomolecular sequences may be considered as a random function from the set of possible phylogenetic trees to the set of collections of biomolecular sequences of observed species. Our resul...