Subdivisions of Transitive Tournaments

We prove that, for r ≥ 2 andn ≥ n(r), every directed graph with n vertices and more edges than the r -partite Turán graph T(r, n) contains a subdivision of the transitive tournament on r + 1 vertices. Furthermore, the extremal graphs are the orientations ofT (r, n) induced by orderings of the vertex classes.     

A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99     

A Note on a Packing Problem in Transitive Tournaments

Let TT_n be a transitive tournament on n vertices. We show that for any directed acyclic graph G of order n and of size not greater than two directed graphs isomorphic to G are arc disjoint subgraphs of TT_n. Moreover, this bound is generally the best possible. The research partially supported by KBN grant 2 P03A 016 18     

The cores of random hypergraphs with a given degree sequence

We study random r‐uniform n vertex hypergraphs with fixed degree sequence d = (d₁…,d_n), maximum degree Δ = o(n^1/24) and total degree θn, where θ is bounded. We give the size, number of edges and degree sequence of the κ ≥ 2) up to a whp error of O(n^2/3 Δ^4/3 log n). In the case of graphs (r = 2) we give further structural details such as the number of tree components and, for the case of smooth degree sequences, the size of the mantle. We give various examples, such as the cores of r‐uniform hypergraphs with a near Poisson degree sequence, and an improved upper bound for the first linear dependence among the columns in the independent column model of random Boolean matrices. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 25, 2004     

Finding Scores in Tournaments

A tournamentT_nis an orientation of the complete graph onnvertices. We continue the algorithmic study initiated by10of recognizing various directed trees in tournaments. Hell and Rosenfeld studied the complexity of finding various oriented paths in tournaments by probing edge directions. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree) in the same model. We show that the complexity of finding a vertex of outdegreek( ≤ (n − 1)/2) inT_nis Θ(nk). This bound is in sharp contrast to the Θ(n) bound for selection in the case of transitive tournaments. We also establish tight bounds for finding vertices of prescribed degree from the adjacency matrix of general directed/undirected graphs. These bounds generalize the classical bound of11for finding a sink (a vertex of outdegree 0 and indegreen − 1) in a directed graph.     

Hit-and-run mixes fast

It is shown that the “hit-and-run” algorithm for sampling from a convex body K (introduced by R.L. Smith) mixes in time O ^*(n ² R ²/r ²), where R and r are the radii of the inscribed and circumscribed balls of K. Thus after appropriate preprocessing, hit-and-run produces an approximately uniformly distributed sample point in time O ^*(n ³), which matches the best known bound for other sampling algorithms. We show that the bound is best possible in terms of R,r and n. Received January 26, 1998 / Revised version received October 26, 1998?Published online July 19, 1999     

New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants

Let G be a finite abelian group (written additively) of rank r with invariants n ₁, n ₂, . . . , n _r , where n _r is the exponent of G. In this paper, we prove an upper bound for the Davenport constant D(G) of G as follows; D(G) ≤ n _r + n _r-1 + (c(3) − 1)n _r-2 + (c(4) − 1) n _r-3 + · · · + (c(r) − 1)n ₁ + 1, where c(i) is the Alon–Dubiner constant, which depends only on the rank of the group \mathbb Z_nrⁱ{{\mathbb Z}_{n_r}^i}. Also, we shall give an application of Davenport’s constant to smooth numbers related to the Quadratic sieve.     

A note on random greedy coloring of uniform hypergraphs

The smallest number of edges forming an n‐uniform hypergraph which is not r‐colorable is denoted by m(n,r). Erd?s and Lovász conjectured that . The best known lower bound was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on the analysis of a random greedy coloring algorithm investigated by Pluhár in 2009. The proof method extends to the case of r‐coloring, and we show that for any fixed r we have improving the bound of Kostochka from 2004. We also derive analogous bounds on minimum edge degree of an n‐uniform hypergraph that is not r‐colorable. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 407–413, 2015     

The cycle structure of two rows in a random Latin square

Let L be chosen uniformly at random from among the latin squares of order n ≥ 4 and let r,s be arbitrary distinct rows of L. We study the distribution of σ_r,s, the permutation of the symbols of L which maps r to s. We show that for any constant c > 0, the following events hold with probability 1 ‐ o(1) as n → ∞: (i) σ_r,s has more than (log n)^1?c cycles, (ii) σ_r,s has fewer than 9 cycles, (iii) L has fewer than n^5/2 intercalates (latin subsquares of order 2). We also show that the probability that σ_r,s is an even permutation lies in an interval and the probability that it has a single cycle lies in [2n^‐2,2n^‐2/3]. Indeed, we show that almost all derangements have similar probability (within a factor of n^3/2) of occurring as σ_r,s as they do if chosen uniformly at random from among all derangements of {1,2,…,n}. We conjecture that σ_r,s shares the asymptotic distribution of a random derangement. Finally, we give computational data on the cycle structure of latin squares of orders n ≤ 11. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Manifolds with small Dirac eigenvalues are nilmanifolds

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar curvature. Let r = 1 if n = 2,3 and r = 2^[n/2]-1 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost non-negative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.      

On the Caccetta–Häggkvist Conjecture

 It was conjectured by Caccetta and H?ggkvist in 1978 that every digraph G with n vertices and minimum outdegree at least r contains a directed cycle of length at most ⌈n/r⌉. By refining an argument of Chvátal and Szemerédi, we prove that such G contains a directed cycle of length at most n/r+73. Received: September 13, 1999 Final version received: June 19, 2000 Acknowledgment. I want to thank a referee for many valuable suggestions leading to the clear presentation of the paper.     

Mukai-Sakai bound for equivariant principal bundles

Mukai and Sakai proved that given a vector bundleE of rankn on a connected smooth projective curve of genusg and anyr∈[1,n], there is subbundleS of rankr such that deg Hom(S, E/S)≤r(n−r)g. We prove a generalization of this bound for equivariant principal bundles. Our proof even simplifies the one given by Holla and Narasimhan for usual principal bundles.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

On the number of cycles possible in digraphs with large girth

We consider directed graphs which have no short cycles. In particular, if n is the number of vertices in a graph which has no cycles of length less than n ? k, for some constant k < ?n, then we show that the graph has no more than 3^k cycles. In addition, we show that for k ≤ ½n, there are graphs with exactly 3^k cycles. We thus are able to show that it is possible to bound the number of cycles possible in a graph which has no cycles of length less than f(n) by a polynomial in n if and only if f(n) ≥ n ? rlog(n) for some r.     

Multivariate Polynomial Minimization and Its Application in Signal Processing

We make a conjecture that the number of isolated local minimum points of a 2n-degree or (2n+1)-degree r-variable polynomial is not greater than n ^r when n 2. We show that this conjecture is the minimal estimate, and is true in several cases. In particular, we show that a cubic polynomial of r variables may have at most one local minimum point though it may have 2^r critical points. We then study the global minimization problem of an even-degree multivariate polynomial whose leading order coefficient tensor is positive definite. We call such a multivariate polynomial a normal multivariate polynomial. By giving a one-variable polynomial majored below a normal multivariate polynomial, we show the existence of a global minimum of a normal multivariate polynomial, and give an upper bound of the norm of the global minimum and a lower bound of the global minimization value. We show that the quartic multivariate polynomial arising from broad-band antenna array signal processing, is a normal polynomial, and give a computable upper bound of the norm of the global minimum and a computable lower bound of the global minimization value of this normal quartic multivariate polynomial. We give some sufficient and necessary conditions for an even order tensor to be positive definite. Several challenging questions remain open.     

On the stationary quasi-Newtonian flow obeying a power-law

We investigate in this paper existence of a weak solution for a stationary incompressible Navier-Stokes system with non-linear viscosity and with non-homogeneous boundary conditions for velocity on the boundary. Our concern is with the viscosity obeying the power-law dependence ν(ξ) = ∣Tr(ξξ*)∣^r/2?1, r < 2, on shear stress ξ. It is corresponding to most quasi-Newtonian flows with injection on the boundary. Since for r ? 2 the inertial term precludes any a priori estimate in general, we suppose the Reynolds number is not too large. Using the specific algebraic structure of the Navier-Stokes system we prove existence of at least one approximate solution. The constructed approximate solution turns out to be uniformly bounded in W^1,r (Omega;)ⁿ and using monotonicity and compactness we successfully pass to the limit for r ≥ 3n/(n + 2). For 3n/(n + 2) > r > 2n/(n + 2) our construction gives existence of at least one very weak solution. Furthermore, for r ≥ 3n/(n + 2) we prove that all weak solutions lying in the ball in W of radius smaller than critical are equal. Finally, we obtain an existence result for the flow through a thin slab.     

Off‐diagonal and asymptotic results on the Ramsey number r(K2,m,K2,n)

An upper bound on the Ramsey number r(K_2,n‐s,K_2,n) where s ≥ 2 is presented. Considering certain r(K_2,n‐s,K_2,n)‐colorings obtained from strongly regular graphs, we additionally prove that this bound matches the exact value of r(K_2,n‐s,K_2,n) in infinitely many cases if holds. Moreover, the asymptotic behavior of r(K_2,m,K_2,n) is studied for n being sufficiently large depending on m. We conclude with a table of all known Ramsey numbers r(K_2,m,K_2,n) where m,n ≤ 10. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 252–268, 2003     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

Tearing a strip off the plane

In this article we examine some homomorphic properties of certain subgraphs of the unit-distance graph. We define G_r to be the subgraph of the unit-distance graph induced by the subset (−∞, ∞) × [0, r] of the plane. The bulk of the article is devoted to examining the graphs G_r, when r is the minimum width such that G_r contains an odd cycle of given length. We determine for each odd n the minimum width r_n such that contains an n-cycle C_n, and characterize the embeddings of C_n in $G_{r_{n}}$. We then show that is homomorphically equivalent to C_n when n ≡ 3 (mod 4), but is a core when n ≡ 1 (mod 4). We begin by showing that G_r is homomorphically compact for each r ≥ 0, as defined in [1]. We conclude with some other interesting results and open problems related to the graphs G_r. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 17–33, 1998     

Online Ramsey games for triangles in random graphs

In the online F-avoidance edge-coloring game with r colors, a graph on n vertices is generated by randomly adding a new edge at each stage. The player must color each new edge as it appears; the goal is to avoid a monochromatic copy of F. Let N₀(F,r,n) be the threshold function for the number of edges that the player is asymptotically almost surely able to paint before he/she loses. Even when F=K₃, the order of magnitude of N₀(F,r,n) is unknown for r≥3. In particular, the only known upper bound is the threshold function for the number of edges in the offline version of the problem, in which an entire random graph on n vertices with m edges is presented to the player to be r edge-colored. We improve the upper bound for the online triangle-avoidance game with r colors, providing the first result that separates the online threshold function from the offline bound for r≥3. This supports a conjecture of Marciniszyn, Spöhel, and Steger that the known lower bound is tight for cliques and cycles for all r.