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.

DOI

https://doi.org/10.57709/1059728

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