Date of Award
Master of Science (MS)
Mathematics and Statistics
We consider tiling problems for graphs and hypergraphs. For two graphs and , an -tiling of is a subgraph of consisting of only vertex disjoint copies of . By using the absorbing method, we give a short proof that in a balanced tripartite graph , if every vertex is adjacent to of the vertices in each of the other vertex partitions, then has a -tiling. Previously, Magyar and Martin  proved the same result (without ) by using the Regularity Lemma.
In a 3-uniform hypergraph , let denote the minimum number of edges that contain for all pairs of vertices. We show that if , there exists a -tiling that misses at most vertices of . On the other hand, we show that there exist hypergraphs such that and does not have a perfect -tiling. These extend the results of Pikhurko  on -tilings.
Lightcap, Andrew, "Minimum Degree Conditions for Tilings in Graphs and Hypergraphs." Thesis, Georgia State University, 2011.