In this thesis, we investigate the problem of packing and covering odd (u, v)-trails in a graph. A (u, v)-trail is a (u, v)-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Given a graph G and two specified vertices u and v, the odd (u, v)-trail packing number, denoted by ν(u, v), is the maximum number of edge-...

A directed covering design, DC(v, k, λ), is a (v, k, 2λ) covering design in which the blocks are regarded as ordered k-tuples and in which each ordered pair of elements occurs in at least λ blocks. Let DE(v, k, λ) denote the minimum number of blocks in a DC(v, k, λ). In this paper the values of the function DE(v, 5, λ) are determined for all even integers v ≥ 5 and λ odd.

In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

Given a set S of v ≥ 2 symbols, and integers k ≥ t ≥ 2 and N ≥ 1, an N × k array A ∈ SN×k is an (N ; t, k, v)-covering array if all sequences in S appear as rows in every N × t subarray of A. These arrays have a wide variety of applications, driving the search for small covering arrays. The covering array number, CAN(t, k, v), is the smallest N for which an (N ; t, k, v)-covering array exists. ...