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Coloring graphs with forbidden minors

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Date Issued:
2017
Abstract/Description:
A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger's Conjecture from 1943 which states that every graph with no Kt-minor is (t - 1)-colorable. This conjecture has been proved true for t ? 6, but remains open for all t ? 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no K_t-minor is (2t - 6)-colorable for t = 7, 8, 9, in the process giving a shorter and computer-free proof of the known results for t = 7, 8. We also show that this result extends to larger values of t if Mader's bound for the extremal function for Kt-minors is true. Additionally, we show that any graph with no K8?-minor is 9-colorable, and any graph with no K8?-minor is 8-colorable. The Kempe-chain method developed for our proofs of the above results may be of independent interest. We also use Mader's H-Wege theorem to establish some sufficient conditions for a graph to contain a K8-minor.Another motivation for my research is a well-known conjecture of Erd?s and Lov(&)#225;sz from 1968, the Double-Critical Graph Conjecture. A connected graph G is double-critical if for all edges xy ? E(G), ?(G - x - y) = ?(G) - 2. Erd?s and Lov(&)#225;sz conjectured that the only double-critical t-chromatic graph is the complete graph Kt. This conjecture has been show to be true for t ? 5 and remains open for t ? 6. It has further been shown that any non-complete, double-critical, t-chromatic graph contains Kt as a minor for t ? 8. We give a shorter proof of this result for t = 7, a computer-free proof for t = 8, and extend the result to show that G contains a K9-minor for all t ? 9. Finally, we show that the Double-Critical Graph Conjecture is true for double-critical graphs with chromatic number t ? 8 if such graphs are claw-free.
Title: Coloring graphs with forbidden minors.
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Name(s): Rolek, Martin, Author
Song, Zixia, Committee Chair
Brennan, Joseph, Committee Member
Reid, Michael, Committee Member
Zhao, Yue, Committee Member
DeMara, Ronald, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2017
Publisher: University of Central Florida
Language(s): English
Abstract/Description: A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger's Conjecture from 1943 which states that every graph with no Kt-minor is (t - 1)-colorable. This conjecture has been proved true for t ? 6, but remains open for all t ? 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no K_t-minor is (2t - 6)-colorable for t = 7, 8, 9, in the process giving a shorter and computer-free proof of the known results for t = 7, 8. We also show that this result extends to larger values of t if Mader's bound for the extremal function for Kt-minors is true. Additionally, we show that any graph with no K8?-minor is 9-colorable, and any graph with no K8?-minor is 8-colorable. The Kempe-chain method developed for our proofs of the above results may be of independent interest. We also use Mader's H-Wege theorem to establish some sufficient conditions for a graph to contain a K8-minor.Another motivation for my research is a well-known conjecture of Erd?s and Lov(&)#225;sz from 1968, the Double-Critical Graph Conjecture. A connected graph G is double-critical if for all edges xy ? E(G), ?(G - x - y) = ?(G) - 2. Erd?s and Lov(&)#225;sz conjectured that the only double-critical t-chromatic graph is the complete graph Kt. This conjecture has been show to be true for t ? 5 and remains open for t ? 6. It has further been shown that any non-complete, double-critical, t-chromatic graph contains Kt as a minor for t ? 8. We give a shorter proof of this result for t = 7, a computer-free proof for t = 8, and extend the result to show that G contains a K9-minor for all t ? 9. Finally, we show that the Double-Critical Graph Conjecture is true for double-critical graphs with chromatic number t ? 8 if such graphs are claw-free.
Identifier: CFE0006649 (IID), ucf:51227 (fedora)
Note(s): 2017-05-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): vertex coloring -- double-critical graphs -- claw-free graphs -- graph minors -- Hadwiger's conjecture
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0006649
Restrictions on Access: public 2017-05-15
Host Institution: UCF

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