Ryan R. Martin

Codes identifying sets of vertices in random networks

In this paper we deal with codes identifying sets of vertices in random networks; that is, (1,=<@?)-identifying codes. These codes enable us to detect sets of faulty processors in a multiprocessor system, assuming that the maximum number of faulty processors is bounded by a fixed constant @?. The (1,=<1)-identifying codes are of special interest. For random graphs we use the model G(n,p), in which each one of the (n2) possible edges exists with probability p. We give upper and lower bounds on the minimum cardinality of a (1,=<@?)-identifying code in a random graph, as well as threshold functions for the property of admitting such a code. We derive existence results from probabilistic constructions. A connection between identifying codes and superimposed codes is also established.

On diamond-free subposets of the Boolean lattice

The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A@?B,C@?D. A diamond-free family in the n-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements B and C may or may not be related. There is a diamond-free family in the n-dimensional Boolean lattice of size (2-o(1))(n@?n/2@?). In this paper, we prove that any diamond-free family in the n-dimensional Boolean lattice has size at most (2.25+o(1))(n@?n/2@?). Furthermore, we show that the so-called Lubell function of a diamond-free family in the n-dimensional Boolean lattice which contains the empty set is at most 2.25+o(1), which is asymptotically best possible.

How many random edges make a dense graph Hamiltonian

This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph Hamiltonian with high probability. Adding Θ(n) random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.

Q 2 -free Families in the Boolean Lattice

For a family

Adding random edges to dense graphs

{{\cal F}}

Expected values of parameters associated with the minimum rank of a graph

of subsets of [n] = {1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that

Induced saturation number

{{\cal F}}

Sub-Ramsey Numbers for Arithmetic Progressions

is P-free if it does not contain a subposet isomorphic to P. Let ex(n, P) be the largest size of a P-free family of subsets of [n]. Let Q2 be the poset with distinct elements a, b, c, d, a < b,c < d; i.e., the 2-dimensional Boolean lattice. We show that 2N − o(N) ≤ ex(n, Q2) ≤ 2.283261N + o(N), where

On the Strong Chromatic Number of Graphs

N = \binom{n}{\lfloor n/2 \rfloor}

On the computation of edit distance functions

. We also prove that the largest Q2-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.