The utilization of partitions is essential for proving the properties of different types of graphs. Gallai-Ramsey problems and conjectures which require the Regularity lemma require unique methods to improve the bounds on known results. In this work the upper bounds for Gallai-Ramsey using $k$ colors is lowered to at most $k(n-1) +3n$ for even cycles and $(2^{k+3}-3)n \\log n$ for odd cycles. Also, with the ideas of partitions in mind, the Regularity lemma was used to show that it is possible to create short paths with a fixed end point in hopes of pursuing the Enomoto and Ota conjecture.
","abstract_html":"<p>The utilization of partitions is essential for proving the properties of different types of graphs. Gallai-Ramsey problems and conjectures which require the Regularity lemma require unique methods to improve the bounds on known results. In this work the upper bounds for Gallai-Ramsey using $k$ colors is lowered to at most $k(n-1) +3n$ for even cycles and (2k+3-3)n \\log n for odd cycles. Also, with the ideas of partitions in mind, the Regularity lemma was used to show that it is possible to create short paths with a fixed end point in hopes of pursuing the Enomoto and Ota conjecture.</p>","abstract_has_math":true,"creators":["Hall, Martin L"],"institution":null,"degree_name":"Master of Science in Mathematics (M.S.)","degree_level":"Thesis (restricted to Georgia Southern)","degree_discipline":"Department of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Andrew Sills","Hua Wang"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-01-01T08:00:00Z","date_published":"2014-01-01T08:00:00Z","updated_at":"2026-07-24T02:27:52Z","subjects":["ETD","Regularity","Gallai-Ramsey","Ramsey","Partitions","Cycles","Paths","Discrete Mathematics and Combinatorics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.georgiasouthern.edu/etd/1047","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Andrew Sills","Hua Wang"]},{"key":"dc:creator","label":"Author","values":["Hall, Martin L"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2014-04-01T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis (restricted to Georgia Southern)"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Mathematics (M.S.)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ETD","Regularity","Gallai-Ramsey","Ramsey","Partitions","Cycles","Paths","Discrete Mathematics and Combinatorics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.georgiasouthern.edu/etd/1047"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The utilization of partitions is essential for proving the properties of different types of graphs. Gallai-Ramsey problems and conjectures which require the Regularity lemma require unique methods to improve the bounds on known results. In this work the upper bounds for Gallai-Ramsey using $k$ colors is lowered to at most $k(n-1) +3n$ for even cycles and $(2^{k+3}-3)n \\log n$ for odd cycles. Also, with the ideas of partitions in mind, the Regularity lemma was used to show that it is possible to create short paths with a fixed end point in hopes of pursuing the Enomoto and Ota conjecture.
"]},{"key":"dc:title","label":"Title","values":["Utilization of Partitions in Graph Structures"]}]}],"canonical_facts":{"dc:contributor":["Andrew Sills","Hua Wang"],"dc:creator":["Hall, Martin L"],"dc:date.available":["2014-04-01T07:00:00Z"],"dc:description.abstract":["The utilization of partitions is essential for proving the properties of different types of graphs. Gallai-Ramsey problems and conjectures which require the Regularity lemma require unique methods to improve the bounds on known results. In this work the upper bounds for Gallai-Ramsey using $k$ colors is lowered to at most $k(n-1) +3n$ for even cycles and $(2^{k+3}-3)n \\log n$ for odd cycles. Also, with the ideas of partitions in mind, the Regularity lemma was used to show that it is possible to create short paths with a fixed end point in hopes of pursuing the Enomoto and Ota conjecture.
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