Mathematics, Department of

Permanent URI for this collectionhttps://hdl.handle.net/1773/53060

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    Graph Genera and Minors
    (2025-12-05) Ulrigg. Austin
    This thesis studies embedding graphs on surfaces, with an emphasis on orientable surfaces, rotation systems, and forbidden minors. We begin by developing the historical background, from the earliest occurrences of the crossing number $\operatorname{cr}(G)$ of a graph, to Heawood's map-coloring conjecture and the eventual invention of rotation systems as a way of encoding cellular embeddings of graphs on surfaces. We then describe and analyze \textsc{PAGE}, a rotation system based genus algorithm developed jointly with Alexander Metzger. For an arbitrary graph $G$ with $n$ vertices and $m$ edges, \textsc{PAGE} determines the orientable genus of $G$ in $\mathcal{O}(n(4^m/n)^{n/t})$ steps where $t$ is the \textit{girth} of $G$. We illustrate \textsc{PAGE} with examples where it was used to determine the genus of graphs whose genus was previously unknown, most notably the $(3,12)$ cage which has genus 17. Additionally, we discuss ways in which \textsc{PAGE} can be improved and other strategies that could be used to determine the genus of graphs such as the $(3,11)$ cage which are so far out of reach for algorithmic approaches. We also give an overview of the problem of determining the full set $\mathfrak{F}(S_1)$ of forbidden toroidal minors, summarize known results, and discuss possible directions for further progress on completing the set.
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    A Practical Algorithmic Approach to Graph Embedding
    (2025-09-03) Metzger
    Minimal-genus graph embedding is about drawing graphs on surfaces with no edges crossing and as few holes as possible. This thesis first covers the necessary background in topological graph theory to understand graph embeddings through rotation systems. It then studies an adaptation of this approach, Practical Algorithm for Graph Embedding (PAGE), that takes advantage of the cycle sequence of a graph to work more efficiently in practice, especially for graphs of high girth or low degree. This enables it to determine the previously intractable genus of the (3, 12)-cage as 17.
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    Stories of Success: Disabled Undergraduate Students' Experiences and Achievements
    (2025-05-20) Beck, Emma; Emery, Jack
    Students with disabilities face unique challenges in higher education, from navigating inaccessible environments to advocating for academic accommodations. This study presents insights from interviews with eight college students from several universities across the United States, with diverse disabilities, highlighting their experiences, obstacles, and strategies for success. Through in-depth conversation, participants shared how they have overcome barriers related to physical accessibility, learning differences, mental health, and social inclusion. Key themes emerged, including the importance of self-advocacy, institutional support, and personal resilience. Many students emphasized the role of disability services, assistive technologies, and supportive faculty in fostering an inclusive learning environment. Others described how their experiences shaped their problem-solving skills, adaptability, and determination—qualities that have contributed to their academic and personal growth. Despite systemic challenges, each participant’s journey reflects the broader potential for success when equitable support structures are in place. These narratives not only highlight the strengths and contributions of students with disabilities but also offer valuable insights for educators, administrators, and policymakers striving to create more inclusive campuses. By amplifying their voices, this study underscores the need for continued efforts in accessibility, awareness, and institutional change.