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On Square Paths
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On Square Paths
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Dissertation
On Square Paths
2026
Overview
In graph theory, a square path is a path in which each vertex is adjacent to the vertices two places ahead and behind. A square cycle is defined analogously. A path or cycle is Hamilton in a graph G if it includes every vertex of G. Pósa conjectured that every graph with minimum degree at least 2/3 its order has a Hamilton square cycle. This dissertation presents a proof of Pósa’s conjecture for graphs on at least 62,000 vertices, an improvement over the previous state of the art. I also explore some auxiliary questions that emerge naturally when exploring square paths. For example, it turns out that there exists a natural analogue of a component, called a triangle component, that seems to describe connectivity by square paths. This dissertation includes some initial results regarding triangle components and suggests avenues for further research. Also, one might seek to rearrange a square path using prefix reversals, and this leads to a novel combinatorial problem concerning prefix reversals of binary and quaternary strings. This dissertation presents a complete result for the binary case and some initial results for the quaternary case.
Publisher
ProQuest Dissertations & Theses
Subject
ISBN
9798277438657
