Combinatorial Generalizations of Sieve Methods and Characterizing Hamiltonicity via Induced Subgraphs
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A sieve method is in effect an application of the inclusion-exclusion counting principle, and the estimation methods to avoid computing the explicit formula. Sieve methods have been used in number theory for over a hundred years. These methods have been modified to make use of the structure of integer-like objects; producing better estimates and providing more use cases. The first part of the thesis aims to analyze and use the analogues of number theoretic sieves in combinatorial contexts. This part consists of my work with Yu-Ru Liu in Chapters 2 and 3. We focus on two sieve methods: the Turán sieve (introduced by Liu and Murty in 2005) and the Selberg sieve (independently generalized by Wilson in 1969 and Chow in 1998 with slightly different formulations). Some comparisons and applications of these sieves are discussed. In particular, we apply the combinatorial Turán sieve to count labelled graphs and we apply the combinatorial Selberg sieve to count subspaces of finite spaces. Finding sufficient conditions for Hamiltonicity in graphs is a classical topic, where the difficulty is bracketed by the NP-hardness of the associated decision problem. The second part of the thesis, consisting of Chapter 4, aims to characterize Hamiltonicity by means of induced subgraphs. The results in this chapter are based on the paper "Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs." Discrete Mathematics, 345(7):112869, 2022, co-authored with Joseph Cheriyan, Sepehr Hajebi, and Sophie Spirkl. We study induced subgraphs and conditions for Hamiltonicity. In particular, we characterize the minimal 2-connected non-Hamiltonian split graphs and the minimal 2-connected non-Hamiltonian triangle-free graphs.
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Zishen Qu (2022). Combinatorial Generalizations of Sieve Methods and Characterizing Hamiltonicity via Induced Subgraphs. UWSpace. http://hdl.handle.net/10012/18552