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Two Questions about Properties of Large Graphs: On Generalized Turán Numbers and the Chromatic Number of Random Lifts

J. D Nir, University of Nebraska - Lincoln

Abstract

In this thesis, I consider questions and results from two fields: extremal graph theory and random graph theory. First, I consider a generalization of the classic theorem of Turán that states that the maximum number of edges in a clique-free graph is uniquely obtained by a balanced complete r-partite graph. Following Alon and Shikhelman, instead of counting edges while forbidding cliques, I count copies of a target graph T while forbidding copies of F. I investigate the behavior of the extremal graphs and their graphon limits in the case where T and F are stars or cliques and provide partial results in more general cases. Second, using the small subgraph conditioning method and several other techniques, I obtain an upper bound for the chromatic number of random lifts of a general d-regular graph. In the case where the host graph G is complete, I provide a nearly matching lower bound to prove that asymptotically almost surely the chromatic number of a random n-lift of a clique takes one of at most two values. This result complements results on the chromatic number of regular graphs chosen uniformly with the configuration model.

Subject Area

Mathematics

Recommended Citation

Nir, J. D, "Two Questions about Properties of Large Graphs: On Generalized Turán Numbers and the Chromatic Number of Random Lifts" (2020). ETD collection for University of Nebraska - Lincoln. AAI27956779.
https://digitalcommons.unl.edu/dissertations/AAI27956779

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