{"id":{"repo_id":"gatech","oai_identifier":"oai:repository.gatech.edu:1853/72694"},"canonical_url":"https://search.dev.ndltd.org/etd/gatech/oai:repository.gatech.edu:1853/72694","repository":{"repo_id":"gatech","name":"Georgia Tech","base_url":"https://repository.gatech.edu/server/oai/request"},"display":{"title":"Two graph classes with bounded chromatic number","abstract":"A class of graphs is said to be $\\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\\chi(G) \\leq f(\\omega(G)$, and polynomially $\\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\\chi$-bounded for any $t$ with binding function $f(\\omega) = O(\\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\\chi$-binding function $f(\\omega) = \\omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.","abstract_html":"A class of graphs is said to be $\\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies <span class=\"etd-inline-math\">\\chi(G) \\leq f(&omega;(G)</span>, and polynomially $\\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\\chi$-bounded for any $t$ with binding function <span class=\"etd-inline-math\">f(&omega;) = O(&omega;<sup>t+1</sup>)</span>. A class of graphs is said to satisfy the Vizing bound if it admits the $\\chi$-binding function <span class=\"etd-inline-math\">f(&omega;) = &omega; + 1</span>. It was conjectured that (fork, <span class=\"etd-inline-math\">K<sub>3</sub></span>)-free graphs would be 3-colorable, where fork is the graph obtained from <span class=\"etd-inline-math\">K<sub>1, 4</sub></span> by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.","abstract_has_math":true,"creators":["Schroeder, Joshua"],"institution":"Georgia Institute of Technology","degree_name":null,"degree_level":"Doctoral","degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Yu, Xingxing"],"committee_chairs":[],"committee_members":["Bernshteyn, Anton","Kelly, Tom","Wang, Zhiyu","Lu, Linyuan"],"year":2023,"date_issued":"2023-07-17","date_published":"2023-07-17","updated_at":"2026-07-27T19:51:09Z","subjects":["Graph theory","graph coloring","chromatic number","chi-boundedness","chi-binding functions"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1853/72694","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Yu, Xingxing"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Bernshteyn, Anton","Kelly, Tom","Wang, Zhiyu","Lu, Linyuan"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Schroeder, Joshua"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2023-09-06T19:48:14Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2023-09-06T19:48:14Z"]},{"key":"dc:date.issued","label":"Date","values":["2023-07-17"]},{"key":"dc:publisher","label":"Institution","values":["Georgia Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Graph theory","graph coloring","chromatic number","chi-boundedness","chi-binding functions"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1853/72694"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["A class of graphs is said to be $\\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\\chi(G) \\leq f(\\omega(G)$, and polynomially $\\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\\chi$-bounded for any $t$ with binding function $f(\\omega) = O(\\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\\chi$-binding function $f(\\omega) = \\omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Two graph classes with bounded chromatic number"]}]}],"canonical_facts":{"dc:contributor.advisor":["Yu, Xingxing"],"dc:contributor.committeemember":["Bernshteyn, Anton","Kelly, Tom","Wang, Zhiyu","Lu, Linyuan"],"dc:contributor.department":["Mathematics"],"dc:creator":["Schroeder, Joshua"],"dc:date.accessioned":["2023-09-06T19:48:14Z"],"dc:date.available":["2023-09-06T19:48:14Z"],"dc:date.issued":["2023-07-17"],"dc:description.abstract":["A class of graphs is said to be $\\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\\chi(G) \\leq f(\\omega(G)$, and polynomially $\\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\\chi$-bounded for any $t$ with binding function $f(\\omega) = O(\\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\\chi$-binding function $f(\\omega) = \\omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample."],"dc:description.degree":["Ph.D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/1853/72694"],"dc:language.iso":["en_US"],"dc:publisher":["Georgia Institute of Technology"],"dc:subject":["Graph theory","graph coloring","chromatic number","chi-boundedness","chi-binding functions"],"dc:title":["Two graph classes with bounded chromatic number"],"dc:type":["Text"],"thesis:degree_level":["Doctoral"]},"updated_at":"2026-07-27T19:51:09Z"}