#### Date of Award

7-13-2009

#### Degree Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematics and Statistics

#### First Advisor

Yi Zhao - Chair

#### Second Advisor

Johannes Hattingh

#### Third Advisor

Guantao Chen

#### Abstract

Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3.

The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.

#### Recommended Citation

Bush, Albert, "Two Problems on Bipartite Graphs." Thesis, Georgia State University, 2009.

https://scholarworks.gsu.edu/math_theses/72