Showing papers in "Electronic Journal of Combinatorics in 2000"
TL;DR: It is proved thatgamma(G) is the domination number of a graph and H is the Cartesian product of graphs for all simple graphs and H.
Abstract: Let $\gamma(G)$ denote the domination number of a graph $G$ and let $G\square H$ denote the Cartesian product of graphs $G$ and $H$. We prove that $\gamma(G)\gamma(H) \le 2 \gamma(G\square H)$ for all simple graphs $G$ and $H$.
89 citations
TL;DR: In this paper, a continued fraction of the number of 132-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $12\dots k was given for Chebyshev polynomials of the second kind.
Abstract: Let $f_n^r(k)$ be the number of 132-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $12\dots k$, and let $F_r(x;k)$ and $F(x,y;k)$ be the generating functions defined by $F_r(x;k)=\sum_{n\ge 0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\ge 0}F_r(x;k)y^r$. We find an explicit expression for $F(x,y;k)$ in the form of a continued fraction. This allows us to express $F_r(x;k)$ for $1\le r\le k$ via Chebyshev polynomials of the second kind.
80 citations
TL;DR: This work construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations using multivariate interpolation polynomials associated with Schur's S and P functions and Jack symmetric functions.
Abstract: We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type integrals.
71 citations
TL;DR: In this paper, the authors prove a conjecture of Cohn and Propp about the symmetry of the set of alternating sign matrices (ASMs) under the dihedral group rearranging those vertices, which is much bigger than the group of symmetries of the square.
Abstract: We prove a conjecture of Cohn and Propp, which refines a conjecture of Bosley and Fidkowski about the symmetry of the set of alternating sign matrices (ASMs) We examine data arising from the representation of an ASM as a collection of paths connecting $2n$ vertices and show it to be invariant under the dihedral group $D_{2n}$ rearranging those vertices, which is much bigger than the group of symmetries of the square We also generalize conjectures of Propp and Wilson relating some of this data for different values of $n$
65 citations
TL;DR: A new numerical procedure for generating dense packings of disks and spheres inside various geometric shapes is described, and it is believed that in some of the smaller cases, these packings are in fact optimal.
Abstract: We describe a new numerical procedure for generating dense packings of disks and spheres inside various geometric shapes. We believe that in some of the smaller cases, these packings are in fact optimal. When applied to the previously studied cases of packing n equal disks in a square, the procedure conrms all the previous record packings [NO1] [NO2] [GL], except for n = 32, 37, 48, and 50 disks, where better packings than those previously recorded are found. For n = 32 and 48, the new packings are minor variations of the previous record packings. However, for n =3 7 and 50, the new patterns dier substantially. For example, they are mirror-symmetric, while the previous record packings are not. AMS subject classication: primary 05B40, secondary 90C59
63 citations
TL;DR: The main technique is the 'step by step' approach of [3], which examines G_p one step at a time in such a way that the dependence on what has gone before can be split into 'positive' and 'negative' parts, using the notions of up-sets and down-sets.
Abstract: Let $\mathcal{Q}$ be a monotone decreasing property of graphs $G$ on $n$ vertices. Erdős, Suen and Winkler [5] introduced the following natural way of choosing a random maximal graph in $\mathcal{Q}$: start with $G$ the empty graph on $n$ vertices. Add edges to $G$ one at a time, each time choosing uniformly from all $e\in G^c$ such that $G+e\in \mathcal{Q}$. Stop when there are no such edges, so the graph $G_\infty$ reached is maximal in $\mathcal{Q}$. Erdős, Suen and Winkler asked how many edges the resulting graph typically has, giving good bounds for $\mathcal{Q}=\{$bipartite graphs$\}$ and $\mathcal{Q}=\{$triangle free graphs$\}$. We answer this question for $C_4$-free graphs and for $K_4$-free graphs, by considering a related question about standard random graphs $G_p\in \mathcal{G}(n,p)$. The main technique we use is the 'step by step' approach of [3]. We wish to show that $G_p$ has a certain property with high probability. For example, for $K_4$ free graphs the property is that every 'large' set $V$ of vertices contains a triangle not sharing an edge with any $K_4$ in $G_p$. We would like to apply a standard Martingale inequality, but the complicated dependence involved is not of the right form. Instead we examine $G_p$ one step at a time in such a way that the dependence on what has gone before can be split into 'positive' and 'negative' parts, using the notions of up-sets and down-sets. The relatively simple positive part is then estimated directly. The much more complicated negative part can simply be ignored, as shown in [3].
61 citations
TL;DR: This paper considers bijections between sets of lattice paths defined on various sets of permitted steps which yield path counts related to the Narayana polynomials.
Abstract: Let $d(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ using the steps (0,1), (1,0), and (1,1). Let $e(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ with permitted steps from the step set ${\bf N} \times {\bf N} - \{(0,0)\}$, where ${\bf N}$ denotes the nonnegative integers. We give a bijective proof of the identity $e(n) = 2^{n-1} d(n)$ for $n \ge 1$. In giving perspective for our proof, we consider bijections between sets of lattice paths defined on various sets of permitted steps which yield path counts related to the Narayana polynomials.
59 citations
TL;DR: This is the first known constructive lower bound of order $d^2$ of order Euclidean-space, and compares with the well known "absolute" upper bound of d(d+1)$ lines in any equiangular set.
Abstract: A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .
55 citations
TL;DR: Six conjectures stating that if a (multi)hypergraph H has $n$ vertices and does not contain $p$ then the size of ${\cal H}$ is $O(n)$ and the number of such ${\ cal H}s is £O(c^n)$.
Abstract: A (multi)hypergraph ${\cal H}$ with vertices in ${\bf N}$ contains a permutation $p=a_1a_2\ldots a_k$ of $1, 2, \ldots, k$ if one can reduce ${\cal H}$ by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to ${\cal H}_p=(\{i, k+a_i\}:\ i=1, \ldots, k)$. We formulate six conjectures stating that if ${\cal H}$ has $n$ vertices and does not contain $p$ then the size of ${\cal H}$ is $O(n)$ and the number of such ${\cal H}$s is $O(c^n)$. The latter part generalizes the Stanley–Wilf conjecture on permutations. Using generalized Davenport–Schinzel sequences, we prove the conjectures with weaker bounds $O(n\beta(n))$ and $O(\beta(n)^n)$, where $\beta(n)\rightarrow\infty$ very slowly. We prove the conjectures fully if $p$ first increases and then decreases or if $p^{-1}$ decreases and then increases. For the cases $p=12$ (noncrossing structures) and $p=21$ (nonnested structures) we give many precise enumerative and extremal results, both for graphs and hypergraphs.
53 citations
TL;DR: Type-B analogues of combinatorial statistics previously studied on noncrossing partitions are defined and it is shown that analogous equidistribution and symmetry properties hold in the case of type-B noncrossed partitions.
Abstract: We define type-B analogues of combinatorial statistics previously studied on noncrossing partitions and show that analogous equidistribution and symmetry properties hold in the case of type-B noncrossing partitions. We also identify pattern-avoiding classes of elements in the hyperoctahedral group which parallel known classes of restricted permutations with respect to their relations to noncrossing partitions.
51 citations
TL;DR: It is constructively proved that the partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain.
Abstract: We constructively prove that the partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain. In other words, there exists an infinite collection of permutations no one of which contains another as a pattern.
TL;DR: New upper bounds for the connective constant of self-avoiding walks in a hypercubic lattice are obtained by automatic generation of finite automata for counting walks with finite memory.
Abstract: New upper bounds for the connective constant of self-avoiding walks in a hypercubic lattice are obtained by automatic generation of finite automata for counting walks with finite memory. The upper bound in dimension two is 2.679192495.
TL;DR: It is proved that every oriented path has oriented game chromatic number at most 7 (and this bound is tight) and that there exists a constant t such thatevery oriented outerplanar graph has orientedGame chromaticNumber at most t.
Abstract: We consider the oriented version of a coloring game introduced by Bodlaender [On the complexity of some coloring games, Internat. J. Found. Comput. Sci. 2 (1991), 133{147]. We prove that every oriented path has oriented game chromatic number at most 7 (and this bound is tight), that every oriented tree has oriented game chromatic number at most 19 and that there exists a constant t such that every oriented outerplanar graph has oriented game chromatic number at most t.
TL;DR: New lower bounds for the Schur numbers S(6) and S(7) are given and several observations concerning symmetric sumfree partitions into 5 sets are made.
Abstract: We give new lower bounds for the Schur numbers S(6) and S(7). This will imply new lower bounds for the Multicolor Ramsey Numbers R6(3) and R7(3). We also make several observations concerning symmetric sumfree partitions into 5 sets.
TL;DR: It is shown, that explicit construction of such low rank matrices imply explicit constructions of Ramsey graphs.
Abstract: We examine nn matrices overZm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n 1=(p 1) O(1) ). If m is a non-prime-power integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c p logn log logn). We also show, that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs.
TL;DR: In this article, a number of new combinatorial operations on skew semistandard domino tableaux were proposed, which complement constructions dened by C. Carr e and B. Leclerc.
Abstract: We dene a number of new combinatorial operations on skew semistandard domino tableaux that complement constructions dened by C. Carr e and B. Leclerc, and clarify the link with ordinary skew semistandard tableaux and the Littlewood-Richardson rule. These operations are: (1) a bijection between semistandard domino tableaux and certain pairs of ordinary tableaux of the same weight that together ll the same shape, and which determine the \plactic class" of the domino tableau; (2) a weight preserving reversible transformation of domino tableaux into ordinary tableaux of a related shape (the correspondence involves 2-quotients) mapping the subset of Yamanouchi domino tableaux onto that of the Littlewood-Richardson tableaux; (3) a correspondence between Yamanouchi domino tableaux of shape and weight and Yamanouchi domino tableaux of shape 0 and weight ,w here 0 is scaled horizontally and vertically by a factor 2 .T he essential properties of (1) and (2) are obtained by proving their commutation with the \coplactic" (or crystal) operations (which for domino tableaux were dened by Carr e and Leclerc). Construction (2) allows algorithmic separation of the Littlewood-Richardson tableaux describing the decomposition of the tensor square of a general linear group representation into contributions to its symmetric and alternating parts.
TL;DR: It is shown that the performance ratio of any online coloring algorithm with advance knowledge of the input graph is at least $\Omega(N/\log^2 N)$, where $N$ is the number of vertices and this matches and generalizes the bound for the case of an unknown input graph.
Abstract: The problem of online coloring an unknown graph is known to be hard. Here we consider the problem of online coloring in the relaxed situation where the input must be isomorphic to a given known graph. All that foils a computationally powerful player is that it is not known to which sections of the graph the vertices to be colored belong. We show that the performance ratio of any online coloring algorithm with advance knowledge of the input graph is at least $\Omega(N/\log^2 N)$, where $N$ is the number of vertices. This matches and generalizes the bound for the case of an unknown input graph. We also show that any online independent set algorithm has a performance ratio at least $N/8$.
TL;DR: It turns out that $s({\cal C))\le m$ is uniquely determined up to isomorphism by the intersection numbers of the scheme $\widehat{\cal C}^{(m)}$ iff ${cal C}, and $t({Âcal C})=1 $ iff $\Gamma$ is distance-transitive.
Abstract: To each coherent configuration (scheme) ${\cal C}$ and positive integer $m$ we associate a natural scheme $\widehat{\cal C}^{(m)}$ on the $m$-fold Cartesian product of the point set of ${\cal C}$ having the same automorphism group as ${\cal C}$. Using this construction we define and study two positive integers: the separability number $s({\cal C})$ and the Schurity number $t({\cal C})$ of ${\cal C}$. It turns out that $s({\cal C})\le m$ iff ${\cal C}$ is uniquely determined up to isomorphism by the intersection numbers of the scheme $\widehat{\cal C}^{(m)}$. Similarly, $t({\cal C})\le m$ iff the diagonal subscheme of $\widehat{\cal C}^{(m)}$ is an orbital one. In particular, if ${\cal C}$ is the scheme of a distance-regular graph $\Gamma$, then $s({\cal C})=1$ iff $\Gamma$ is uniquely determined by its parameters whereas $t({\cal C})=1$ iff $\Gamma$ is distance-transitive. We show that if ${\cal C}$ is a Johnson, Hamming or Grassmann scheme, then $s({\cal C})\le 2$ and $t({\cal C})=1$. Moreover, we find the exact values of $s({\cal C})$ and $t({\cal C})$ for the scheme ${\cal C}$ associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular, $s({\cal C})=t({\cal C})=2$ if ${\cal C}$ is the scheme of a Doob graph. In addition, we prove that $s({\cal C})\le 2$ and $t({\cal C})\le 2$ for any imprimitive 3/2-homogeneous scheme. Finally, we show that $s({\cal C})\le 4$, whenever ${\cal C}$ is a cyclotomic scheme on a prime number of points.
TL;DR: This paper combines their technique with concepts introduced by several authors in a series of papers on game chromatic number to show that for every positive integer k,t here exists an integer t so that if C is a topologically closed class of graphs and C does not contain a complete graph on k vertices, then whenever G is an orientation of a graph from C, the oriented game Chromatic number of G is at most t.
Abstract: Ne set ril and Sopena introduced a concept of oriented game chromatic number and developed a general technique for bounding this parameter. In this paper, we combine their technique with concepts introduced by several authors in a series of papers on game chromatic number to show that for every positive integer k ,t here exists an integer t so that if C is a topologically closed class of graphs and C does not contain a complete graph on k vertices, then whenever G is an orientation of a graph from C, the oriented game chromatic number of G is at most t .I n particular, oriented planar graphs have bounded oriented game chromatic number. This answers a question raised by Ne set ril and Sopena. We also answer a second question raised by Ne set ril and Sopena by constructing a family of oriented graphs for which oriented game chromatic number is bounded but extended Go number is not.
TL;DR: A surprising construction is provided which shows equality only for certain classes modulo 4 and a matrix to be simple if it is a (0,1)-matrix with no repeated columns is defined.
Abstract: The present paper continues the work begun by Anstee, Ferguson, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let $F$ be a $k\times l$ (0,1)-matrix (the forbidden configuration). Assume $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define ${\hbox{forb}}(m,F)$ as the largest $n$, which would depend on $m$ and $F$, so that such an $A$ exists. 'Small' refers to the size of $k$ and in this paper $k=2$. For $p\le q$, we set $F_{pq}$ to be the $2\times (p+q)$ matrix with $p$ $\bigl[{1\atop0}\bigr]$'s and $q$ $\bigl[{0\atop1}\bigr]$'s. We give new exact values: ${\hbox{forb}}(m,F_{0,4})=\lfloor {5m\over2}\rfloor +2$, ${\hbox{forb}}(m,F_{1,4})=\lfloor {11m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{1,5})=\lfloor {15m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{2,4})=\lfloor {10m\over3}-{4\over3}\rfloor$ and ${\hbox{forb}}(m,F_{2,5})=4m$ (For ${\hbox{forb}}(m,F_{1,4})$, ${\hbox{forb}}(m,F_{1,5})$ we obtain equality only for certain classes modulo 4). In addition we provide a surprising construction which shows ${\hbox{forb}}(m,F_{pq})\ge \bigl({p+q\over2}+O(1)\bigr)m$.
TL;DR: In this paper, a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels is given. But this is not the same as the result of Robertson, Wilf and Zeilberger.
Abstract: We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of $(132)$-pattern avoiding permutations that have a given number of increasing patterns of length $k$. This extends and illuminates a result of Robertson, Wilf and Zeilberger for the case $k=3$.
TL;DR: It is shown that a simply chosen random linear permutation will suffice for an average set from the point of view of approximate min-wise independence as $k,p \to\infty$, ${\bf E}_X[F(X)]={{1}\over {k}}+O\left({(\log k)^3}\ over {k^{3/2}}\right)$
Abstract: A set of permutations ${\cal F} \subseteq S_n$ is min-wise independent if for any set $X \subseteq [n]$ and any $x \in X$, when $\pi$ is chosen at random in ${\cal F}$ we have ${\bf P} \left(\min\{\pi(X)\} = \pi(x)\right) = {{1}\over {|X|}}$. This notion was introduced by Broder, Charikar, Frieze and Mitzenmacher and is motivated by an algorithm for filtering near-duplicate web documents. Linear permutations are an important class of permutations. Let $p$ be a (large) prime and let ${\cal F}_p=\{p_{a,b}:\;1\leq a\leq p-1,\,0\leq b\leq p-1\}$ where for $x\in [p]=\{0,1,\ldots,p-1\}$, $p_{a,b}(x)=ax+b\pmod p$. For $X\subseteq [p]$ we let $F(X)=\max_{x\in X}\left\{{\bf P}_{a,b}(\min\{p(X)\}=p(x))\right\}$ where ${\bf P}_{a,b}$ is over $p$ chosen uniformly at random from ${\cal F}_p$. We show that as $k,p \to\infty$, ${\bf E}_X[F(X)]={{1}\over {k}}+O\left({(\log k)^3}\over {k^{3/2}}\right)$ confirming that a simply chosen random linear permutation will suffice for an average set from the point of view of approximate min-wise independence.
TL;DR: It is shown that maximal planar bi-hypergraphs are 2-colourable, formulas for their chromatic polynomial and chromatic spectrum in terms of 2-factors in the dual are found and it is proved that their Chromatic spectrum is gap-free and a sharp estimate is provided on the maximum number of colours in a colouring.
Abstract: A mixed hypergraph is a triple ${\cal H} = (V,{\cal C}, {\cal D})\;$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, the ${\cal C}$-edges and ${\cal D}$-edges, respectively. A $k$-colouring of ${\cal H}$ is a mapping $c: V\rightarrow [k]$ such that each ${\cal C}$-edge has at least two vertices with a ${\cal C}$ommon colour and each ${\cal D}$-edge has at least two vertices of ${\cal D}$ifferent colours. ${\cal H}$ is called a planar mixed hypergraph if its bipartite representation is a planar graph. Classic graphs are the special case of mixed hypergraphs when ${\cal C}=\emptyset$ and all the ${\cal D}$-edges have size 2, whereas in a bi-hypergraph ${\cal C} = {\cal D}$. We investigate the colouring properties of planar mixed hypergraphs. Specifically, we show that maximal planar bi-hypergraphs are 2-colourable, find formulas for their chromatic polynomial and chromatic spectrum in terms of 2-factors in the dual, prove that their chromatic spectrum is gap-free and provide a sharp estimate on the maximum number of colours in a colouring.
TL;DR: If G is sufficiently highly edge connected then the expected length of a minimum spanning tree is at most $\sim {n\over r}\zeta(3)$ and if the authors omit the edge connectivity condition, then it is at least at most $n\ over r}(\zeta (3)+1)$.
Abstract: Consider a connected $r$-regular $n$-vertex graph $G$ with random independent edge lengths, each uniformly distributed on $[0,1]$. Let $mst(G)$ be the expected length of a minimum spanning tree. We show in this paper that if $G$ is sufficiently highly edge connected then the expected length of a minimum spanning tree is $\sim {n\over r}\zeta(3)$. If we omit the edge connectivity condition, then it is at most $\sim {n\over r}(\zeta(3)+1)$.
TL;DR: It is shown that that the authors can have at most $\log p$ values of $q$ which give a linear f(n,p,q) and that this behavior is similar to that of the linear and quadratic orders of magnitude.
Abstract: For fixed integers $p$ and $q$, an edge coloring of $K_n$ is called a $(p, q)$-coloring if the edges of $K_n$ in every subset of $p$ vertices are colored with at least $q$ distinct colors. Let $f(n, p, q)$ be the smallest number of colors needed for a $(p, q)$-coloring of $K_n$. In [3] Erdős and Gyarfas studied this function if $p$ and $q$ are fixed and $n$ tends to infinity. They determined for every $p$ the smallest $q$ ($= {p \choose 2} - p + 3$) for which $f(n,p,q)$ is linear in $n$ and the smallest $q$ for which $f(n,p,q)$ is quadratic in $n$. They raised the question whether perhaps this is the only $q$ value which results in a linear $f(n,p,q)$. In this paper we study the behavior of $f(n,p,q)$ between the linear and quadratic orders of magnitude. In particular we show that that we can have at most $\log p$ values of $q$ which give a linear $f(n,p,q)$.
TL;DR: The main result of this paper is a characterisation of the abstract nite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral.
Abstract: Two tournaments T1 and T2 on the same vertex set X are said to be switching equivalent if X has a subset Y such that T2 arises from T1 by switching all arcs between Y and its complement XnY . The main result of this paper is a characterisation of the abstract nite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral. Moreover, if G is such a group, then there is a switching class C ,w ith Aut(C) = G, such that every subgroup ofG of odd order is the full automorphism group of some tournament in C. Unlike previous results of this type, we do not give an explicit construction, but only an existence proof. The proof follows as a special case of a result on the full automorphism group of randomG-invariant digraphs selected from a certain class of probability distributions. We also show that a permutation group G, acting on a set X ,i s contained in the automorphism group of some switching class of tournaments with vertex set X if and only if the Sylow 2-subgroups of
TL;DR: The probability that the finest coarsening of all p_i$'s is the one-block partition is shown to approach $1$ for every r, and the lattice of the set partitions ordered by refinement is studied.
Abstract: The lattice of the set partitions of $[n]$ ordered by refinement is studied. Suppose $r$ partitions $p_1,\dots,p_r$ are chosen independently and uniformly at random. The probability that the coarsest refinement of all $p_i$'s is the finest partition $\bigl\{\{1\},\dots,\{n\}\bigr\}$ is shown to approach $0$ for $r=2$, and $1$ for $r\ge 3$. The probability that the finest coarsening of all $p_i$'s is the one-block partition is shown to approach $1$ for every $r\ge 2$.
TL;DR: It is convenient to distinguish three cases depending on the nature of the power series for the structures: purely formal, convergent on the circle of convergence, and other.
Abstract: Let $\rho _n$ be the fraction of structures of "size" $n$ which are "connected"; e.g., (a) the fraction of labeled or unlabeled $n$-vertex graphs having one component, (b) the fraction of partitions of $n$ or of an $n$-set having a single part or block, or (c) the fraction of $n$-vertex forests that contain only one tree. Various authors have considered $\lim \rho _n$, provided it exists. It is convenient to distinguish three cases depending on the nature of the power series for the structures: purely formal, convergent on the circle of convergence, and other. We determine all possible values for the pair $(\liminf \rho _{n},\;\limsup \rho _{n})$ in these cases. Only in the convergent case can one have $0
TL;DR: An ILP-formulation based on a new class of inequalities (subtour number constraints) is presented and the associated Hamiltonian $p–median polytope is examined, in particular its dimension and its affine hull.
Abstract: We deal, from a theoretical point of view, with the asymmetric Hamiltonian $p$–median problem. This problem, which has many applications, can be viewed as a mixed routing location problem. An ILP-formulation based on a new class of inequalities (subtour number constraints) is presented. The associated Hamiltonian $p$–median polytope is examined, in particular its dimension and its affine hull. We determine which of the defining inequalities induce facets.
TL;DR: A sharp estimate is given for the number of distinct sums a+b with $(a,b)
otin\ {\cal R}$, and a partial generalization of this estimate is obtained for arbitrary Abelian groups.
Abstract: In 1980, Erdős and Heilbronn posed the problem of estimating (from below) the number of sums $a+b$ where $a\in A$ and $b\in B$ range over given sets $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ of residues modulo a prime $p$, so that $a
eq b$. A solution was given in 1994 by Dias da Silva and Hamidoune. In 1995, Alon, Nathanson and Ruzsa developed a polynomial method that allows one to handle restrictions of the type $f(a,b)
eq 0$, where $f$ is a polynomial in two variables over ${\Bbb Z}/p{\Bbb Z}$. In this paper we consider restricting conditions of general type and investigate groups, distinct from ${\Bbb Z}/p{\Bbb Z}$. In particular, for $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ and ${\cal R}\subseteq A\times B$ of given cardinalities we give a sharp estimate for the number of distinct sums $a+b$ with $(a,b)
otin\ {\cal R}$, and we obtain a partial generalization of this estimate for arbitrary Abelian groups.