Extremal problems involving forbidden subgraphs

In this thesis, we study extremal problems involving forbidden subgraphs. We are interested in extremal problems over a family of graphs or over a family of hypergraphs. In Chapter 2, we consider improper coloring of graphs without short cycles. We find how sparse an improperly critical graph can be when it has no short cycle. In particular, we find the exact threshold of density of triangle-free $(0,k)$-colorable graphs and we find the asymptotic threshold of density of $(j,k)$-colorable graphs of large girth when $k\geq 2j+2$. In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of $r$-dynamic chromatic number of graphs in terms of other parameters. In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs. In particular, we characterize hypergraphs with the maximum number of edges that contain no $r$-regular subgraphs. We also establish upper bounds for the number of edges in graphs and hypergraphs with no edge-disjoint equicovering subgraphs.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Kim, Jaehoon

Contributors dc:contributor

• Kostochka, Alexandr V.
• Weichsel, Paul M.
• West, Douglas B.
• Lidicky, Bernard

• Extremal Graph Theory
• Graphs
• Graph Coloring
• Hypergraphs
• Forbidden Subgraphs

Statement dc:rights

• Copyright 2014 Jaehoon Kim

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/49480

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/49480