Completing Partial Packings of Bipartite Graphs

Given a bipartite graph $H$ and an integer $n$, let $f(n;H)$ be the smallest integer such that, any set of edge disjoint copies of $H$ on $n$ vertices, can be extended to an $H$-design on at most $n+f(n;H)$ vertices. We establish tight bounds for the growth of $f(n;H)$ as $n \rightarrow \infty$. In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that $f(n;H) = o(n)$. This settles a long-standing open problem

New Bounds for Permutation Codes in Ulam Metric

New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.Comment: To be presented at ISIT 2015, 5 page

On some batch code properties of the simplex code

The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a simple, constructive proof of a result that is "in between" these two properties. Our approach is to relate these properties to certain (old and new) additive problems in finite abelian groups. We also formulate a conjecture for finite abelian groups that generalizes the above-mentioned conjecture

On Integer Sequences, Packings and Games on Graphs

This dissertation concerns four problems in combinatorics. In Chapter 2 we consider the Prolonger-Shortener game of F saturation, introduced by Füredi, Reimer and Seress: Players take turns drawing edges on an initially edgeless vertex set of size n with the restriction that they do not complete a copy of a graph in F. The game ends when no more edges can be drawn. Prolonger wants as many edges as possible at the end of the game and Shortener as few as possible. We ask what is the final number of edges with both players playing optimally when F is a fixed path, or a collection of trees. We also consider a directed version of the game. Chapter 3 concerns completion of partial packings of copies of a bipartite graph H. An H-design on n vertices is an edge-disjoint collection of copies of H whose edge sets partition the edge set of the complete graph on n vertices. Given a bipartite graph H and an integer n, let f(n;H) be the smallest integer such that any set of edge-disjoint copies of H on n vertices can be extended to an H-design on at most n+f(n;H) vertices. We establish tight bounds for the growth of f(n;H) as n approaches infinity. In particular, we prove the conjecture of Füredi and Lehel that f(n;H)=o(n). Chapter 4 is dedicated to a particular integer sequence, the Slowgrow sequence, originally introduced by Steven Kalikow. It starts with 1, and having defined terms s1,...,sn the term sn+1 is the smallest positive integer m such that the block sn-m+2...sn+1 has not occurred in the sequence earlier. Our main result is that blocks which can potentially occur multiple times in the sequence actually occur infinitely often. We also prove bounds on the time of the first occurrence of n in the Slowgrow sequence and that the limiting density of every number in the sequence is 0. Chapter 5 is motivated by a question of András Sárközy. We prove sufficient conditions for existence of infinite sets of natural numbers A and B such that the number of solutions of the equation a+b=n where a is in A and b is in B is monotone increasing for n\u3en0. We also examine a generalized notion of Sidon sets, that is, sets A, B with the property that, for every n\u3e=0, the equation above has at most one solution, i.e., all pairwise sums are distinct

Maximal sets of k-spaces pairwise intersecting in at least a (k-2)-space

In this paper, we analyze the structure of maximal sets of k-dimensional spaces in PG(n, q) pairwise intersecting in at least a (k - 2)-dimensional space, for 3 <= k <= n - 2. We give an overview of the largest examples of these sets with size more than f(k, q) = max{3q(4) + 6q(3) + 5q(2) + q + 1,theta(k+1) + q(4) + 2q(3) + 3q(2)}