{"id":{"repo_id":"unlv","oai_identifier":"oai:oasis.library.unlv.edu:rtds-2408"},"canonical_url":"https://search.dev.ndltd.org/etd/unlv/oai:oasis.library.unlv.edu:rtds-2408","repository":{"repo_id":"unlv","name":"University of Nevada - Las Vegas","base_url":"https://oasis.library.unlv.edu/do/oai/"},"display":{"title":"From equi-graphical sets to graphical permutations: A problem of degrees in graphs","abstract":"A set {a1, a2,.., an} of positive integers with a 1 < a2 < &cdots; < an is said to be equi-graphical if there exists a graph with exactly ai vertices of degree ai for each i with 1 ≤ i ≤ n. It is known that such a set is equi-graphical if and only if i=1nai is even and an≤i=1n-1 ai2 . This concept is now generalized to the following problem: Given a set S of positive integers and a permutation pi on S, determine when there exists a graph containing exactly ai vertices of degree pi(ai) for each i (1 ≤ i ≤ n). If such a graph exists, then pi is called a graphical permutation; In this paper, the graphical permutations on sets of size four are characterized and using a criterion of Fulkerson, Hoffman, and McAndrew, we show that a permutation pi of S = {a1, a2, .. , an}, where 1 ≤ a1 < a2 < &cdots; < an and such that pi(a n) = an, is graphical if and only if i=1naip ai is even and an≤i=1n-1 aipai .","abstract_html":"A set {a1, a2,.., an} of positive integers with a 1 < a2 < &cdots; < an is said to be equi-graphical if there exists a graph with exactly ai vertices of degree ai for each i with 1 &le; i &le; n. It is known that such a set is equi-graphical if and only if i=1nai is even and an&le;i=1n-1 ai2 . This concept is now generalized to the following problem: Given a set S of positive integers and a permutation pi on S, determine when there exists a graph containing exactly ai vertices of degree pi(ai) for each i (1 &le; i &le; n). If such a graph exists, then pi is called a graphical permutation; In this paper, the graphical permutations on sets of size four are characterized and using a criterion of Fulkerson, Hoffman, and McAndrew, we show that a permutation pi of S = {a1, a2, .. , an}, where 1 &le; a1 < a2 < &cdots; < an and such that pi(a n) = an, is graphical if and only if i=1naip ai is even and an&le;i=1n-1 aipai .","abstract_has_math":false,"creators":["Watson, Michael Lee"],"institution":"University of Nevada, Las Vegas","degree_name":"Master of Science (MS)","degree_level":"Thesis","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":["Michelle Schultz"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2002,"date_issued":"2002-01-01T08:00:00Z","date_published":"2002-01-01T08:00:00Z","updated_at":"2026-07-24T05:25:33Z","subjects":[],"languages":["English"],"rights":["IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://oasis.library.unlv.edu/rtds/1409"],"render_values":[{"text":"https://oasis.library.unlv.edu/rtds/1409","href":"https://oasis.library.unlv.edu/rtds/1409","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.25669/naxf-z507","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Michelle Schultz"]},{"key":"dc:creator","label":"Author","values":["Watson, Michael Lee"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["University of Nevada, Las Vegas"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["10.25669/naxf-z507","https://oasis.library.unlv.edu/rtds/1409","https://oasis.library.unlv.edu/context/rtds/article/2408/viewcontent/uc.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["A set {a1, a2,.., an} of positive integers with a 1 < a2 < &cdots; < an is said to be equi-graphical if there exists a graph with exactly ai vertices of degree ai for each i with 1 ≤ i ≤ n. It is known that such a set is equi-graphical if and only if i=1nai is even and an≤i=1n-1 ai2 . This concept is now generalized to the following problem: Given a set S of positive integers and a permutation pi on S, determine when there exists a graph containing exactly ai vertices of degree pi(ai) for each i (1 ≤ i ≤ n). If such a graph exists, then pi is called a graphical permutation; In this paper, the graphical permutations on sets of size four are characterized and using a criterion of Fulkerson, Hoffman, and McAndrew, we show that a permutation pi of S = {a1, a2, .. , an}, where 1 ≤ a1 < a2 < &cdots; < an and such that pi(a n) = an, is graphical if and only if i=1naip ai is even and an≤i=1n-1 aipai ."]},{"key":"dc:format","label":"Dc Format","values":["pdf"]},{"key":"dc:title","label":"Title","values":["From equi-graphical sets to graphical permutations: A problem of degrees in graphs"]}]}],"canonical_facts":{"dc:contributor":["Michelle Schultz"],"dc:creator":["Watson, Michael Lee"],"dc:description.abstract":["A set {a1, a2,.., an} of positive integers with a 1 < a2 < &cdots; < an is said to be equi-graphical if there exists a graph with exactly ai vertices of degree ai for each i with 1 ≤ i ≤ n. It is known that such a set is equi-graphical if and only if i=1nai is even and an≤i=1n-1 ai2 . This concept is now generalized to the following problem: Given a set S of positive integers and a permutation pi on S, determine when there exists a graph containing exactly ai vertices of degree pi(ai) for each i (1 ≤ i ≤ n). If such a graph exists, then pi is called a graphical permutation; In this paper, the graphical permutations on sets of size four are characterized and using a criterion of Fulkerson, Hoffman, and McAndrew, we show that a permutation pi of S = {a1, a2, .. , an}, where 1 ≤ a1 < a2 < &cdots; < an and such that pi(a n) = an, is graphical if and only if i=1naip ai is even and an≤i=1n-1 aipai ."],"dc:format":["pdf"],"dc:identifier":["10.25669/naxf-z507","https://oasis.library.unlv.edu/rtds/1409","https://oasis.library.unlv.edu/context/rtds/article/2408/viewcontent/uc.pdf"],"dc:language":["English"],"dc:publisher":["University of Nevada, Las Vegas"],"dc:rights":["IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["From equi-graphical sets to graphical permutations: A problem of degrees in graphs"],"dc:type":["Text"],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:25:33Z"}