Problems in extremal graph theory

We consider a variety of problems in extremal graph and set theory. The {\em chromatic number} of $G$, $\chi(G)$, is the smallest integer $k$ such that $G$ is $k$-colorable. The {\it square} of $G$, written G², is the supergraph of $G$ in which also vertices within distance 2 of each other in $G$ are adjacent. A graph $H$ is a {\it minor} of $G$ if $H$ can be obtained from a subgraph of $G$ by contracting edges. We show that the upper bound for \chi(G²) conjectured by Wegner (1977) for planar graphs holds when $G$ is a K₄-minor-free graph. We also show that \chi(G²) is equal to the bound only when G² contains a complete graph of that order. One of the central problems of extremal hypergraph theory is finding the maximum number of edges in a hypergraph that does not contain a specific forbidden structure. We consider as a forbidden structure a fixed number of members that have empty common intersection as well as small union. We obtain a sharp upper bound on the size of uniform hypergraphs that do not contain this structure, when the number of vertices is sufficiently large. Our result is strong enough to imply the same sharp upper bound for several other interesting forbidden structures such as the so-called strong simplices and clusters. The {\em $n$-dimensional hypercube}, Q_n, is the graph whose vertex set is \{0,1\}ⁿ and whose edge set consists of the vertex pairs differing in exactly one coordinate. The generalized Tur\'an problem asks for the maximum number of edges in a subgraph of a graph $G$ that does not contain a forbidden subgraph $H$. We consider the Tur\'an problem where $G$ is Q_n and $H$ is a cycle of length $4k+2$ with $k\geq 3$. Confirming a conjecture of Erd{\H o}s (1984), we show that the ratio of the size of such a subgraph of Q_n over the number of edges of Q_n is $o(1)$, i.e. in the limit this ratio approaches 0 as $n$ approaches infinity.

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

2010

Author dc:creator

• Ozkahya, Lale

Contributors dc:contributor

• Furedi, Zoltan
• West, Douglas B.
• Kostochka, Alexandr V.
• Vijay, Sujith

• squares of graphs
• Turan problem
• hypercube
• hypergraph
• cluster

Statement dc:rights

• Copyright 2010 Lale Ozkahya

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/16851

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/16851