arXiv subjects

Bill Kay

Publications and source records attributed to Bill Kay.

27 records · Page 2Linked to original sources

arXiv2016

The chromatic number of finite type-graphs

By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]=\{1,2,\ldots, n\}$ for some integers $n\ge k\ge 1$, and in which two such sets are adjacent if and only if they realise a certain order type specified in advance. Examples of such graphs have been investigated in a great variety of contexts in the literature with particular attention being paid to their chromatic number. In recent joint work with Tomasz Łuczak, two of the authors embarked on a systematic study of the chromatic numbers of such type-graphs, formulated a general conjecture determining this number up to a multiplicative factor, and proved various results of this kind. In this article we fully prove this conjecture.

math.CO↗

arXiv2016

The Minimum Number of Edges in Uniform Hypergraphs with Property O

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

math.CO↗

arXiv2013

Bounds on the Maximum Number of Minimum Dominating Sets

We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number γ, of dominating sets of size γ. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all sets of size γ. At the same time, we use a modified adjacency matrix to obtain lower bounds on the number of sets of a given size that do not dominate a graph on n vertices

math.CO↗

arXiv2013

Contributions to the theory of de Bruijn cycles

A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so that each object appears exactly once as a set of consecutive elements in the cycle. In this paper, we show the power of de Bruijn's original theorem, namely that the cycles bearing his name exist for n-letter words on a k-letter alphabet for all values of k,n, to prove that we can create de Bruijn cycles for the assignment of elements of [n]={1,2,....,n} to the sets in any labeled subposet of the Boolean lattice; de Bruijn's theorem corresponds to the case when the subposet in question consists of a single ground element. The landmark work of Chung, Diaconis, and Graham extended the agenda of finding de Bruijn cycles to possibly the next most natural set of combinatorial objects, namely k-subsets of [n]. In this area, important contributions have been those of Hurlbert and Rudoy. Here we follow the direction of Blanca and Godbole, who proved that, in a suitable encoding, de Bruijn cycles can be created for the subsets of [n$ of size in the interval [s,t]; 0<=s<t<=n$. In this paper we generalize this result to exhibit existence of de Bruijn cycles for words with weight between s and t, where these parameters are suitably restricted.

math.CO↗

arXiv2012

Graph Odometry

We address problem of determining edge weights on a graph using non-backtracking closed walks from a vertex. We show that the weights of all of the edges can be determined from any starting vertex exactly when the graph has minimum degree at least three. We also determine the minimum number of walks required to reveal all edge weights.

math.CO↗

arXiv2012

Covering n-Permutations with (n+1)-Permutations

Let S_n be the set of all permutations on [n]:={1,2,....,n}. We denote by kappa_n the smallest cardinality of a subset A of S_{n+1} that "covers" S_n, in the sense that each pi in S_n may be found as an order-isomorphic subsequence of some pi' in A. What are general upper bounds on kappa_n? If we randomly select nu_n elements of S_{n+1}, when does the probability that they cover S_n transition from 0 to 1? Can we provide a fine-magnification analysis that provides the "probability of coverage" when nu_n is around the level given by the phase transition? In this paper we answer these questions and raise others.

math.CO↗

arXiv2009

On Universal Cycles of Labeled Graphs

A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops, graphs with multiple edges (with up to m duplications of each edge), directed graphs, hypergraphs, and k-uniform hypergraphs.

math.CO↗

arXiv2008

Contributions to Seymour's Second Neighborhood Conjecture

Let D be a simple digraph without loops or digons. For any v in V(D) let N_1(v) be the set of all nodes at out-distance 1 from v and let N_2(v) be the set of all nodes at out-distance 2. We provide sufficient conditions under which there must exist some v in V(D) such that |N_1(v)| is less than or equal to |N_2(v)|, as well as examine properties of a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex.

math.CO↗

arXiv2008

Elementary Techniques for Erdos-Ko-Rado-like Theorems

The well-known Erdos-Ko-Rado Theorem states that if F is a family of k-element subsets of {1,2,...,n} (n>2k-1) such that every pair of elements in F has a nonempty intersection, then |F| is at most $\binom{n-1}{k-1}$. The theorem also provides necessary and sufficient conditions for attaining the maximum. We present elementary methods for deriving generalizations of the Erdos-Ko-Rado Theorem on several classes of combinatorial objects. We also extend our results to systems under Hamming intersection.

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