{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/44320"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/44320","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On vertex degrees, graph decomposition, and circular chromatic Ramsey number","abstract":"In this thesis, we study extremal problems about vertex degrees and a variant of Ramsey number of graphs, and also structural problems about graph decomposition. In a list (d_1,...,d_n) of positive integers, let r and s denote the largest and smallest entries. A list is gap-free if each integer between r and s is present. In Chapter 2, we prove that a gap-free list with even sum is graphic if it has at least r+(r+s+1)/(2s) terms. With no restriction on gaps, length at least (r+s+1)^2/(4s) suffices, as proved by Zverovich and Zverovich. Both bounds are sharp within 1. When the gaps between consecutive terms are bounded by g, we prove a more general length threshold that includes both of these results. As a tool, we prove that if a positive list d with even sum has no repeated entries other than r and s (and the length exceeds r), then to prove that d is graphic it suffices to check only the ℓth Erdős--Gallai inequality, where ℓ=max{k: d_k≥k} For outerplanar graphs on n vertices, we determine the maximum number of vertices of degree at least k. For k=4 (and n≥7), the answer is n-4. For k=5 (and n≥4), the answer is ⌊(2n-8)/3⌋ (except one less when n≡1 mod 6). For k≥6 (and n≥k+2), the answer is ⌊(n-6)/(k-4)⌋. As a tool, we determine the maximum sum of the degrees of s vertices. We also determine the maximum sum of the degrees of the vertices with degree at least k. A T-decomposition of a graph G is a decomposition of G into isomorphic copies of T. Let T be a tree with m edges. In Chapter 3, we extend the ideas of Snevily and Avgustinovitch to prove the existence of T-decompositions for more 2m-regular graphs and m-regular bipartite graphs. In particular, for r_1,...,r_k with ∑_{i=1}^k r_i=m, we seek sufficient conditions for every cartesian product of graphs G_1,...,G_k with G_i being 2r_i-regular for all i to have a T-decomposition. One sufficient condition is the existence of a k-edge-coloring of T with r_i edges of color i such that every path in T uses some color once or twice. Another sufficient condition is that r_i≤⌈(m+1)/2⌉ for all i and m/k<4 Finally, in Chapter 4, we introduce the circular chromatic Ramsey number R_{χ_c}(F,G) as the infimum of the circular chromatic numbers χ_c(H) of graphs H such that every red/blue edge-coloring of H yields a red copy of F or a blue copy of G. We prove R_{χ_c}(K_3,K_3)=6 and R_{χ_c}(K_3,K_4)=9. Also, if 2<χ_c(G)≤5/2, then R_{χ_c}(G,G)=4. Furthermore, no graph has circular chromatic Ramsey number between 4 and 5. Also, with R_{χ_c}(z)=inf{R_{χ_c}(G): χ_c(G)≥z}, we prove R_{χ_c}(k)≤k(k-1) for k∈ℕ-{1}.","abstract_html":"In this thesis, we study extremal problems about vertex degrees and a variant of Ramsey number of graphs, and also structural problems about graph decomposition. In a list (d_1,...,d_n) of positive integers, let r and s denote the largest and smallest entries. A list is gap-free if each integer between r and s is present. In Chapter 2, we prove that a gap-free list with even sum is graphic if it has at least r+(r+s+1)/(2s) terms. With no restriction on gaps, length at least (r+s+1)^2/(4s) suffices, as proved by Zverovich and Zverovich. Both bounds are sharp within 1. When the gaps between consecutive terms are bounded by g, we prove a more general length threshold that includes both of these results. As a tool, we prove that if a positive list d with even sum has no repeated entries other than r and s (and the length exceeds r), then to prove that d is graphic it suffices to check only the ℓth Erdős--Gallai inequality, where ℓ=max{k: d_k≥k} For outerplanar graphs on n vertices, we determine the maximum number of vertices of degree at least k. For k=4 (and n≥7), the answer is n-4. For k=5 (and n≥4), the answer is ⌊(2n-8)/3⌋ (except one less when n≡1 mod 6). For k≥6 (and n≥k+2), the answer is ⌊(n-6)/(k-4)⌋. As a tool, we determine the maximum sum of the degrees of s vertices. We also determine the maximum sum of the degrees of the vertices with degree at least k. A T-decomposition of a graph G is a decomposition of G into isomorphic copies of T. Let T be a tree with m edges. In Chapter 3, we extend the ideas of Snevily and Avgustinovitch to prove the existence of T-decompositions for more 2m-regular graphs and m-regular bipartite graphs. In particular, for r_1,...,r_k with ∑_{i=1}^k r_i=m, we seek sufficient conditions for every cartesian product of graphs G_1,...,G_k with G_i being 2r_i-regular for all i to have a T-decomposition. One sufficient condition is the existence of a k-edge-coloring of T with r_i edges of color i such that every path in T uses some color once or twice. Another sufficient condition is that r_i≤⌈(m+1)/2⌉ for all i and m/k&lt;4 Finally, in Chapter 4, we introduce the circular chromatic Ramsey number R_{χ_c}(F,G) as the infimum of the circular chromatic numbers χ_c(H) of graphs H such that every red/blue edge-coloring of H yields a red copy of F or a blue copy of G. We prove R_{χ_c}(K_3,K_3)=6 and R_{χ_c}(K_3,K_4)=9. Also, if 2&lt;χ_c(G)≤5/2, then R_{χ_c}(G,G)=4. Furthermore, no graph has circular chromatic Ramsey number between 4 and 5. Also, with R_{χ_c}(z)=inf{R_{χ_c}(G): χ_c(G)≥z}, we prove R_{χ_c}(k)≤k(k-1) for k∈ℕ-{1}.","abstract_has_math":false,"creators":["Jao, Fang-Kai"],"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.","Kostochka, Alexandr V.","Balogh, József"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-24T22:07:39Z","date_published":"2013-05-24T22:07:39Z","updated_at":"2026-07-22T22:25:34Z","subjects":["Vertex degree","graph realization","graph decomposition","tree decomposition","Graham--Haggkvist conjecture","circular chromatic Ramsey number","chromatic Ramsey number","parameter Ramsey number"],"languages":["en"],"rights":["Copyright 2013 Fang-Kai Jao"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/44320","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["West, Douglas B.","Kostochka, Alexandr V.","Balogh, József"]},{"key":"dc:creator","label":"Author","values":["Jao, Fang-Kai"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-05-24T22:07:39Z","2013-05"]},{"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":["Vertex degree","graph realization","graph decomposition","tree decomposition","Graham--Haggkvist conjecture","circular chromatic Ramsey number","chromatic Ramsey number","parameter Ramsey number"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2013 Fang-Kai Jao"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/44320"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, we study extremal problems about vertex degrees and a variant of Ramsey number of graphs, and also structural problems about graph decomposition. In a list (d_1,...,d_n) of positive integers, let r and s denote the largest and smallest entries. A list is gap-free if each integer between r and s is present. In Chapter 2, we prove that a gap-free list with even sum is graphic if it has at least r+(r+s+1)/(2s) terms. With no restriction on gaps, length at least (r+s+1)^2/(4s) suffices, as proved by Zverovich and Zverovich. Both bounds are sharp within 1. When the gaps between consecutive terms are bounded by g, we prove a more general length threshold that includes both of these results. As a tool, we prove that if a positive list d with even sum has no repeated entries other than r and s (and the length exceeds r), then to prove that d is graphic it suffices to check only the ℓth Erdős--Gallai inequality, where ℓ=max{k: d_k≥k} For outerplanar graphs on n vertices, we determine the maximum number of vertices of degree at least k. For k=4 (and n≥7), the answer is n-4. For k=5 (and n≥4), the answer is ⌊(2n-8)/3⌋ (except one less when n≡1 mod 6). For k≥6 (and n≥k+2), the answer is ⌊(n-6)/(k-4)⌋. As a tool, we determine the maximum sum of the degrees of s vertices. We also determine the maximum sum of the degrees of the vertices with degree at least k. A T-decomposition of a graph G is a decomposition of G into isomorphic copies of T. Let T be a tree with m edges. In Chapter 3, we extend the ideas of Snevily and Avgustinovitch to prove the existence of T-decompositions for more 2m-regular graphs and m-regular bipartite graphs. In particular, for r_1,...,r_k with ∑_{i=1}^k r_i=m, we seek sufficient conditions for every cartesian product of graphs G_1,...,G_k with G_i being 2r_i-regular for all i to have a T-decomposition. One sufficient condition is the existence of a k-edge-coloring of T with r_i edges of color i such that every path in T uses some color once or twice. Another sufficient condition is that r_i≤⌈(m+1)/2⌉ for all i and m/k<4 Finally, in Chapter 4, we introduce the circular chromatic Ramsey number R_{χ_c}(F,G) as the infimum of the circular chromatic numbers χ_c(H) of graphs H such that every red/blue edge-coloring of H yields a red copy of F or a blue copy of G. We prove R_{χ_c}(K_3,K_3)=6 and R_{χ_c}(K_3,K_4)=9. Also, if 2<χ_c(G)≤5/2, then R_{χ_c}(G,G)=4. Furthermore, no graph has circular chromatic Ramsey number between 4 and 5. Also, with R_{χ_c}(z)=inf{R_{χ_c}(G): χ_c(G)≥z}, we prove R_{χ_c}(k)≤k(k-1) for k∈ℕ-{1}.","Item withdrawn by Alexis Thompson (athmpsn1@illinois.edu) on 2013-04-17T17:53:53Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 kylejao-thesis.tex: 234530 bytes, checksum: 5851acdcf6485f30dc720e230f2d6e47 (MD5) Jao_Fang-Kai.pdf: 478878 bytes, checksum: abe5fe7ec503147414132b9b6c673b24 (MD5)","Made available in DSpace on 2013-05-24T22:07:39Z (GMT). No. of bitstreams: 3 Fang-Kai_Jao.pdf: 478878 bytes, checksum: abe5fe7ec503147414132b9b6c673b24 (MD5) kylejao-thesis.tex: 234530 bytes, checksum: 5851acdcf6485f30dc720e230f2d6e47 (MD5) license.txt: 4059 bytes, checksum: 55ad8804d7aa6d85a028c52d41c8e588 (MD5)"]},{"key":"dc:title","label":"Title","values":["On vertex degrees, graph decomposition, and circular chromatic Ramsey number"]}]}],"canonical_facts":{"dc:contributor":["West, Douglas B.","Kostochka, Alexandr V.","Balogh, József"],"dc:creator":["Jao, Fang-Kai"],"dc:date":["2013-05-24T22:07:39Z","2013-05"],"dc:description":["In this thesis, we study extremal problems about vertex degrees and a variant of Ramsey number of graphs, and also structural problems about graph decomposition. In a list (d_1,...,d_n) of positive integers, let r and s denote the largest and smallest entries. A list is gap-free if each integer between r and s is present. In Chapter 2, we prove that a gap-free list with even sum is graphic if it has at least r+(r+s+1)/(2s) terms. With no restriction on gaps, length at least (r+s+1)^2/(4s) suffices, as proved by Zverovich and Zverovich. Both bounds are sharp within 1. When the gaps between consecutive terms are bounded by g, we prove a more general length threshold that includes both of these results. As a tool, we prove that if a positive list d with even sum has no repeated entries other than r and s (and the length exceeds r), then to prove that d is graphic it suffices to check only the ℓth Erdős--Gallai inequality, where ℓ=max{k: d_k≥k} For outerplanar graphs on n vertices, we determine the maximum number of vertices of degree at least k. For k=4 (and n≥7), the answer is n-4. For k=5 (and n≥4), the answer is ⌊(2n-8)/3⌋ (except one less when n≡1 mod 6). For k≥6 (and n≥k+2), the answer is ⌊(n-6)/(k-4)⌋. As a tool, we determine the maximum sum of the degrees of s vertices. We also determine the maximum sum of the degrees of the vertices with degree at least k. A T-decomposition of a graph G is a decomposition of G into isomorphic copies of T. Let T be a tree with m edges. In Chapter 3, we extend the ideas of Snevily and Avgustinovitch to prove the existence of T-decompositions for more 2m-regular graphs and m-regular bipartite graphs. In particular, for r_1,...,r_k with ∑_{i=1}^k r_i=m, we seek sufficient conditions for every cartesian product of graphs G_1,...,G_k with G_i being 2r_i-regular for all i to have a T-decomposition. One sufficient condition is the existence of a k-edge-coloring of T with r_i edges of color i such that every path in T uses some color once or twice. Another sufficient condition is that r_i≤⌈(m+1)/2⌉ for all i and m/k<4 Finally, in Chapter 4, we introduce the circular chromatic Ramsey number R_{χ_c}(F,G) as the infimum of the circular chromatic numbers χ_c(H) of graphs H such that every red/blue edge-coloring of H yields a red copy of F or a blue copy of G. We prove R_{χ_c}(K_3,K_3)=6 and R_{χ_c}(K_3,K_4)=9. Also, if 2<χ_c(G)≤5/2, then R_{χ_c}(G,G)=4. Furthermore, no graph has circular chromatic Ramsey number between 4 and 5. Also, with R_{χ_c}(z)=inf{R_{χ_c}(G): χ_c(G)≥z}, we prove R_{χ_c}(k)≤k(k-1) for k∈ℕ-{1}.","Item withdrawn by Alexis Thompson (athmpsn1@illinois.edu) on 2013-04-17T17:53:53Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 kylejao-thesis.tex: 234530 bytes, checksum: 5851acdcf6485f30dc720e230f2d6e47 (MD5) Jao_Fang-Kai.pdf: 478878 bytes, checksum: abe5fe7ec503147414132b9b6c673b24 (MD5)","Made available in DSpace on 2013-05-24T22:07:39Z (GMT). No. of bitstreams: 3 Fang-Kai_Jao.pdf: 478878 bytes, checksum: abe5fe7ec503147414132b9b6c673b24 (MD5) kylejao-thesis.tex: 234530 bytes, checksum: 5851acdcf6485f30dc720e230f2d6e47 (MD5) license.txt: 4059 bytes, checksum: 55ad8804d7aa6d85a028c52d41c8e588 (MD5)"],"dc:identifier":["http://hdl.handle.net/2142/44320"],"dc:language":["en"],"dc:rights":["Copyright 2013 Fang-Kai Jao"],"dc:subject":["Vertex degree","graph realization","graph decomposition","tree decomposition","Graham--Haggkvist conjecture","circular chromatic Ramsey number","chromatic Ramsey number","parameter Ramsey number"],"dc:title":["On vertex degrees, graph decomposition, and circular chromatic Ramsey number"],"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":"2026-07-22T22:25:34Z"}