{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22749"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22749","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Topics in extremal graph theory","abstract":"New results are proved on several problems in extremal graph theory.","abstract_html":"New results are proved on several problems in extremal graph theory.","abstract_has_math":false,"creators":["Chung, Myung Sook"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["West, Douglas B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:50:11Z","date_published":"2011-05-07T13:50:11Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1993 Chung, Myung Sook"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9411593","(UMI)AAI9411593"],"render_values":[{"text":"AAI9411593","href":null,"code":true},{"text":"(UMI)AAI9411593","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22749","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["West, Douglas B."]},{"key":"dc:creator","label":"Author","values":["Chung, Myung Sook"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:50:11Z","10000-01-01","1993"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1993 Chung, Myung Sook"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9411593","(UMI)AAI9411593","http://hdl.handle.net/2142/22749"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["New results are proved on several problems in extremal graph theory.","Let $ex\\sp*(D;H)$ denote the maximum number of edges in a connected graph with maximum degree D and no induced subgraph isomorphic to the graph H. It is shown that this is finite if and only if H is a disjoint union of paths. Several specific forbidden subgraphs H have been studied, and the following results have been proved:","(1) $ ex\\sp*(D;P\\sb4) = D\\sp2$ for all D, uniquely achieved by $K\\sb{D,D}.$ If, in addition, the maximum clique size is $\\omega$, then the number of edges is at most $D\\sp2 - {D(\\omega-2)\\over 2}.$","(2) $ex\\sp*(D;P\\sb5) = {2\\over 27}D\\sp3 + O(D\\sp2).$","(3) $ex\\sp*(D;2P\\sb3) = {1\\over 8}D\\sp4 +{1\\over 8}D\\sp3 + O(D\\sp2).$","(4) $ex\\sp*(D;P\\sb3 + P\\sb2)< 2D\\sp2.$ If $K\\sb3$ is also forbidden, then $ex\\sp*(D;P\\sb3 + P\\sb2, K\\sb3) = {5\\over 4}D\\sp2 + O(D).$","The p-intersection number of a graph G, denoted by $\\theta\\sb{p}(G),$ is the minimum size of a set $\\cup\\sb{v\\in V(G)}S\\sb{v}$ such that u and v are adjacent if and only if $\\vert S\\sb{u}\\cup S\\sb{v}\\vert \\ge p.$ It is proved here that $\\theta\\sb{p}(K\\sb{n,n})\\ge (n\\sp2 + (2p - 1)n)/p$ for $p\\ge 2.$ Furthermore, $\\theta\\sb2(K\\sb{n,n}) = (n\\sp2 + 3n)/2$ is achieved using a graph design called orthogonal double covering. For sufficiently large p, the residual intersection number, denoted by $\\theta\\sp*(G),$ is defined and studied here as the limiting value of $f\\sb{p}(G) = \\theta\\sb{p}(G)-p.$ The maximum values of n such that $\\theta\\sp*(K\\sb{2,n}) = 5,6$ and 7 are 4, 7, and 14, respectively. Asymptotically, $\\theta\\sp*(K\\sb{2,n}) = \\log\\sb2 n + o(\\log\\sb2 n).$","An $(n,m,r)$-rainbow-free coloring is a multi-edge-coloring of edges in $K\\sb{n}$ with at most m colors such that the edges of each color form a clique and it is not possible to choose distinct colors for each edge in any r-cycle. The maximum value of the sum of the numbers of colors appearing on each edge over all $(n,m,r)$-rainbow-free colorings, denoted by $e(n,m,r),$ was originally investigated by S. Roman. The bounds he demonstrated have been improved upon in this thesis. It is shown that $e(n,m,3) = 2{n-1\\choose 2} + m - 1,$ and that $e(n,m,4)\\le 3{n\\choose 2} + m.$","Made available in DSpace on 2011-05-07T13:50:11Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9411593.pdf: 3581412 bytes, checksum: e982f29369a84e1f1ee2a0521b670a7d (MD5) Previous issue date: 1993","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:59:44Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:12-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Topics in extremal graph theory"]}]}],"canonical_facts":{"dc:contributor":["West, Douglas B."],"dc:creator":["Chung, Myung Sook"],"dc:date":["2011-05-07T13:50:11Z","10000-01-01","1993"],"dc:description":["New results are proved on several problems in extremal graph theory.","Let $ex\\sp*(D;H)$ denote the maximum number of edges in a connected graph with maximum degree D and no induced subgraph isomorphic to the graph H. It is shown that this is finite if and only if H is a disjoint union of paths. Several specific forbidden subgraphs H have been studied, and the following results have been proved:","(1) $ ex\\sp*(D;P\\sb4) = D\\sp2$ for all D, uniquely achieved by $K\\sb{D,D}.$ If, in addition, the maximum clique size is $\\omega$, then the number of edges is at most $D\\sp2 - {D(\\omega-2)\\over 2}.$","(2) $ex\\sp*(D;P\\sb5) = {2\\over 27}D\\sp3 + O(D\\sp2).$","(3) $ex\\sp*(D;2P\\sb3) = {1\\over 8}D\\sp4 +{1\\over 8}D\\sp3 + O(D\\sp2).$","(4) $ex\\sp*(D;P\\sb3 + P\\sb2)< 2D\\sp2.$ If $K\\sb3$ is also forbidden, then $ex\\sp*(D;P\\sb3 + P\\sb2, K\\sb3) = {5\\over 4}D\\sp2 + O(D).$","The p-intersection number of a graph G, denoted by $\\theta\\sb{p}(G),$ is the minimum size of a set $\\cup\\sb{v\\in V(G)}S\\sb{v}$ such that u and v are adjacent if and only if $\\vert S\\sb{u}\\cup S\\sb{v}\\vert \\ge p.$ It is proved here that $\\theta\\sb{p}(K\\sb{n,n})\\ge (n\\sp2 + (2p - 1)n)/p$ for $p\\ge 2.$ Furthermore, $\\theta\\sb2(K\\sb{n,n}) = (n\\sp2 + 3n)/2$ is achieved using a graph design called orthogonal double covering. For sufficiently large p, the residual intersection number, denoted by $\\theta\\sp*(G),$ is defined and studied here as the limiting value of $f\\sb{p}(G) = \\theta\\sb{p}(G)-p.$ The maximum values of n such that $\\theta\\sp*(K\\sb{2,n}) = 5,6$ and 7 are 4, 7, and 14, respectively. Asymptotically, $\\theta\\sp*(K\\sb{2,n}) = \\log\\sb2 n + o(\\log\\sb2 n).$","An $(n,m,r)$-rainbow-free coloring is a multi-edge-coloring of edges in $K\\sb{n}$ with at most m colors such that the edges of each color form a clique and it is not possible to choose distinct colors for each edge in any r-cycle. The maximum value of the sum of the numbers of colors appearing on each edge over all $(n,m,r)$-rainbow-free colorings, denoted by $e(n,m,r),$ was originally investigated by S. Roman. The bounds he demonstrated have been improved upon in this thesis. It is shown that $e(n,m,3) = 2{n-1\\choose 2} + m - 1,$ and that $e(n,m,4)\\le 3{n\\choose 2} + m.$","Made available in DSpace on 2011-05-07T13:50:11Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9411593.pdf: 3581412 bytes, checksum: e982f29369a84e1f1ee2a0521b670a7d (MD5) Previous issue date: 1993","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:59:44Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:12-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9411593","(UMI)AAI9411593","http://hdl.handle.net/2142/22749"],"dc:language":["eng"],"dc:rights":["Copyright 1993 Chung, Myung Sook"],"dc:subject":["Mathematics"],"dc:title":["Topics in extremal graph theory"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:20Z"}