Trees in Connected Graphs

The focus of the Master’s Thesis will be the investigation of current research involving trees that cover subsets of the vertex set of a connected graph. The primary goal is the extension of some recent results and a conjecture of Horak and McAvaney. Given certain conditions, we will reformulate their conjecture that states that if a graph can be spanned by a number of edge-disjoint trees, we can provide a bound on the maximum degree of this collection of edge-disjoint trees. We are able to show that this conjecture is true if the number of trees used to span the graph is one. We will then look at a specific class of graphs, namely series-parallel graphs, and present several new extremal examples to show that these ”tree-like” graphs are difficult to analyze. A comprehensive survey of related literature is also included

On the largest tree of given maximum degree in a connected graph

We prove that every connected graph G contains a tree T of maximum degree at most k that either spans G or has order at least k(G) + 1, where (G) is the minimum degree of G. This generalizes and unifies earlier results of Bermond [1] and Win [7]. We also show that the square of a connected graph contains a spanning tree of maximum degree at most three