Abstract
This dissertation consists of two closely related problems in extremal graph theory.The first problem is a partial resolution of Wang’s conjecture that graphs of girth at least
8 must admit a 2−packing. The work closely follows Wang’s techniques in his proof that
a girth of 12 is sufficient, and utilizes a generalization of the notion of feasible paths to
extend the result to graphs of girth 10. The second problem builds on the unpublished
work of Sean Haler to establish the existence of a 2−packing for all tripartitions of trees
and for all tripartite graphs of order n and size n − 1 without trivial components.