Showing papers in "Contributions to Discrete Mathematics in 2010"
TL;DR: In this article, it was shown that any subset of such a solvable group of small doubling is controlled by a set whose iterated products grow polynomially, and which are contained inside a virtually nilpotent group.
Abstract: Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa to arbitrary abelian groups, where the controlling object is now a coset progression. We extend these results further to solvable groups of bounded derived length, in which the coset progressions are replaced by the more complicated notion of a ``coset nilprogression''. As one consequence of this result, any subset of such a solvable group of small doubling is is controlled by a set whose iterated products grow polynomially, and which are contained inside a virtually nilpotent group. As another application we establish a strengthening of the Milnor-Wolf theorem that all solvable groups of polynomial growth are virtually nilpotent, in which only one large ball needs to be of polynomial size. This result complements recent work of Breulliard-Green, Fisher-Katz-Peng, and Sanders.
55 citations
TL;DR: This paper completely characterize (for both relations) the graphs which force bounded cop number in the classes of graphs defined by forbidding one or more graphs as either sub graphs or induced subgraphs and bound theCop number in terms of the tree-width.
Abstract: The two-player, complete information game of Cops and Robber is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph.
In this paper, we study the cop number in the classes of graphs defined by forbidding one or more graphs as either subgraphs or induced subgraphs. In the case of a single forbidden graph we completely characterize (for both relations) the graphs which force bounded cop number.
In closing, we bound the cop number in terms of the tree-width.
37 citations
TL;DR: A sufficient condition for a generic framework G(p) to be unique is presented and conjecture that this condition is also necessary, and connections with the closely related problems of global rigidity and dimensional rigidity are discussed.
Abstract: Let $V=\{1,\ldots,n\}$ be a finite set. An $r$-configuration is a mapping $p:V \rightarrow R^r$, where
$p^1,\ldots,p^n$ are not contained in a proper hyper-plane. A framework $G(p)$ in $R^r$ is an $r$-configuration together with a graph $G=(V,E)$ such that every two points corresponding to adjacent vertices of $G$ are constrained to stay the same distance apart. A framework $G(p)$ is said to be generic if all the coordinates of $p^1,\ldots, p^n$ are algebraically independent over the integers. A framework $G(p)$ in $R^r$ is said to be unique if there does not exist a framework $G(q)$ in $R^s$, for some $s$, $1 \leq s \leq n-1$, such that $||q^i-q^j||=||p^i-p^j||$ for all $(i,j) \in E$. In this paper we present a sufficient condition for a generic framework $G(p)$ to be unique, and we conjecture that this condition is also necessary. Connections with the closely related problems of global rigidity and dimensional rigidity are also discussed.
34 citations
TL;DR: In this article, certain of the sporadic simple groups are analysed, and the polytopes having these groups as automorphism groups are characterised.
Abstract: In this article, certain of the sporadic simple groups are analysed, and the polytopes
having these groups as automorphism groups are characterised. The sporadic groups considered
include all with order less than 4030387201, that is, all up to and including the order of the Held group.
Four of these simple groups yield no polytopes, and the highest ranked polytopes
are four rank 5 polytopes each from the Higman-Sims group, and the Mathieu group $M_{24}$.
32 citations
TL;DR: A symmetry-adapted notion of `generic' frameworks is introduced with respect to this classification and it is shown that `almost all' realizations in a given symmetry class are generic and all generic realizationsIn this class share the same infinitesimal rigidity properties.
Abstract: In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be embedded injectively in the space. In particular, we introduce a symmetry-adapted notion of `generic' frameworks with respect to this classification and show that `almost all' realizations in a given symmetry class are generic and all generic realizations in this class share the same infinitesimal rigidity properties.
Within this classification we also clarify under what conditions group representation theory techniques can be applied to further analyze the rigidity properties of a (not necessarily injective) symmetric realization.
20 citations
TL;DR: A technique is developed for construction of highly symmetric polytopes using oriented labeled graphs that are permutations representations for groups of automorphisms of chiral or orientably regularpolytopes.
Abstract: We give conditions for oriented labeled graphs that must be satisfied in
order that such are permutations representations for groups of
automorphisms of chiral or orientably regular polytopes. We develop a
technique for construction of highly symmetric polytopes using such
graphs. In particular, we construct chiral polytopes with the
automorphism group $S_n$ for each $n>5$, an infinite family of finite chiral
polytopes of rank $4$, a polytope of rank $5$, as well as several infinite chiral polyhedra.
13 citations
TL;DR: An inductive method of detecting the contractibility of the complex or the dimension of the sphere to which the complex is homotopy equivalent to a sphere is provided.
Abstract: Ehrenborg and Hetyei proved that the independence complex of an arbitrary forest is either contractible or homotopy equivalent to a sphere. The present paper provides an inductive method of detecting the contractibility of the complex or the dimension of the sphere to which the complex is homotopy equivalent.
10 citations
TL;DR: The existence of an analogue of Gallai's theorem relating maximum-size signed b-matchings and minimum-sizesigned b-edge covers for the complete graphs and complete bipartite graphs is proved.
Abstract: In this paper, we study the signed b-edge cover number and the signed b-matching number of a graph. Sharp bounds
on these parameters of the strong product graphs are presented. We prove the existence of an analogue of Gallai's theorem relating maximum-size signed b-matchings and minimum-size signed b-edge covers for the complete graphs and complete bipartite graphs.
8 citations
TL;DR: One family of deletion constructions is presented, which are several methods of generation of new geometric configurations by the judicious deletion of certain points and lines, and introduction of other lines or points.
Abstract: By deletion constructions we mean several methods of generation of new geometric configurations by the judicious deletion of certain points and lines, and introduction of other lines or points. A number of such procedures have recently been developed in a systematic way. We present here one family of such constructions, and will describe other families in the following parts.
8 citations
TL;DR: In this article, a special type of integer partitions in which the parts of the form $p^aq^b$ for some relatively prime integers $p$ and $q$, are restricted by divisibility conditions are considered.
Abstract: We consider a special type of integer partitions in which the parts of the form $p^aq^b$, for some relatively prime integers $p$ and $q$, are restricted by divisibility conditions. We investigate the problems of generating and encoding those partitions and give some estimates for several partition functions.
5 citations
TL;DR: A new random construction is supply for generating infinite families of vertex-transitive tournaments satisfying the adjacency property and it is demonstrated that there is a unique minimum order of $3$-e.
Abstract: Few families of tournaments satisfying the $n$-e.c.\ adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive $n$-e.c.\ tournaments by considering circulant tournaments. Switching is used to generate exponentially many $n$-e.c.\ tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order $3$-e.c.\ tournament of order $19,$ and there are no $3$-e.c.\ tournaments of orders $20,$ $21,$ and $22.$
TL;DR: It is proved that a graphic matroid, M, is isomorphic to a cycle matroid M(G) if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if andonly if $G$ is disjoint union of cacti.
Abstract: Let $M = (E, \mathcal{F})$ be a matroid on a set $E$ and $B$ one of its bases A closed set $\theta \subseteq E$ is saturated with respect to $B$ when $|\theta \cap B | \leq r(\theta)$, where $r(\theta)$ is the rank of $\theta$ The collection of subsets $I$ of $E$ such that $| I \cap \theta| \leq r(\theta)$ for every closed saturated set $\theta$ turns out to be the family of independent sets of a new matroid on $E$, called base-matroid and denoted by $M_B$ In this paper we prove that a graphic matroid $M$, isomorphic to a cycle matroid $M(G)$, is isomorphic to $M_B$, for every base $B$ of $M$, if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if and only if $G$ is disjoint union of cacti Moreover we characterize simple binary matroids $M$ isomorphic to $M_B$, with respect to an assigned base $B$
TL;DR: The second minimal property of the Blaschke-Lebesgue theorem is extended to all normed planes, known as Chakerian’s theorem, and some corollaries from this generalization are given.
Abstract: Among all bodies of constant width in the Euclidean plane, a Reuleaux triangle of width $\lambda$ has minimal area. But Reuleaux triangles are also minimal in another sense: if a convex body can be covered by a translate of any Reuleaux triangle, then it can be covered by a translate of any convex body of the same constant width. The first result is known as the Blaschke-Lebesgue theorem, and it is extended to an arbitrary normed plane in [19] and [5]. In the present paper we extend the second minimal property, known as Chakerian’s theorem, to all normed planes. Some corollaries from this generalization are also given.
TL;DR: This paper deals with the parities of p(n,3) and p (n, 3) and the number of relatively prime partitions into exactly k parts is denoted by p_{\Psi}( n,k).
Abstract: A partition of $n$ is \emph{relatively prime} if its parts form a relatively prime set.
The number of partitions of $n$ into exactly $k$ parts is denoted by $p(n,k)$ and the number
of relatively prime partitions into exactly $k$ parts is denoted by $p_{\Psi}(n,k)$.
In this paper we deal with the parities of $p(n,3)$ and $p_{\Psi}(n,3)$.
TL;DR: In the general case, it is proved that $\beta (V_1,\ldots , V_n)\geq 0$ and the formula yields a formula expressing the length of members of an interval algebra as well as extensions to pseudo-tree algebras.
Abstract: Let $G:=(V,E)$ be a simple graph; for $I\subseteq V$ we denote by
$l(I)$ the number of components of $G[I]$, the subgraph of $G$
induced by $I$. For $V_1,\ldots , V_n$ subsets of $V$, we define a
function $\beta (V_1,\ldots , V_n)$ which is expressed in terms of
$l\left(\bigcup _{i=1} ^{n} V_i\right)$ and $l(V_i\cup
V_j)$ for $i\leq j$. If $V_1,\ldots , V_n$ are pairwise disjoint
independent subsets of $V$, the number $\beta (V_1,\ldots , V_n)$
can be computed in terms of the cyclomatic numbers of
$G\left[\bigcup _{i=1} ^{n} V_i\right]$ and $G[
V_i\cup V_j]$ for $i
eq j$. In the general case, we prove that
$\beta (V_1,\ldots , V_n)\geq 0$ and characterize when $\beta
(V_1,\ldots , V_n)= 0$. This special case yields a formula
expressing the length of members of an interval algebra \cite{s} as
well as extensions to pseudo-tree algebras. Other examples are
given.
TL;DR: This work provides characterizations of tcdd-critical graphs for the classes of block graphs, split graphs and unicyclic graphs and a characterization of cdd- critical cacti.
Abstract: A dominating set in a graph G=(V(G),E(G)) is a set D of vertices
such that every vertex in V(G)\ D has a neighbor in D. A
connected dominating set of a graph G is a dominating set whose
induce subgraph is connected. The connected domination number
gamma_c(G) is the minimum number of vertices of a connected
dominating set of G. A graph G is connected domination
dot-critical (cdd-critical) if contracting any two adjacent vertices
decreases gamma_c(G); and G is totally connected domination
dot-critical (tcdd-critical) if contracting any two vertices decreases
gamma_c(G). We provide characterizations of tcdd-critical graphs
for the classes of block graphs, split graphs and unicyclic graphs and
a characterization of cdd-critical cacti.
TL;DR: For an arbitrary tree, a result is proved about the nature of the least-squares approximation and some properties of the incidence matrix of all the paths in the tree are proved and used.
Abstract: Let T be a tree with vertex set V (T ) = {1, . . . , n}, and with a positive weight associated with each edge. The tree distance between i and j is the weight of the (ij)-path. Given a symmetric, positive real valued function on V (T )×V (T ), we consider the problem of approximating it by a tree distance corresponding to T, by the least-squares method. The problem is solved explicitly when T is a path or a double-star. For an arbitrary tree, a result is proved about the nature of the least-squares approximation. Some properties of the incidence matrix of all the paths in the tree are proved and used.