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 Title: Extremal problems involving forbidden subgraphs Author(s): Kim, Jaehoon Director of Research: Kostochka, Alexandr V. Doctoral Committee Chair(s): Weichsel, Paul M. Doctoral Committee Member(s): Kostochka, Alexandr V.; West, Douglas B.; Lidicky, Bernard Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Extremal Graph Theory Graphs Graph Coloring Hypergraphs Forbidden Subgraphs Abstract: 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. Issue Date: 2014-05-30 URI: http://hdl.handle.net/2142/49480 Rights Information: Copyright 2014 Jaehoon Kim Date Available in IDEALS: 2014-05-30 Date Deposited: 2014-05
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