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Coloring graphs with forbidden minors
 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 Ktminor 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 K7minor is 7colorable. We begin by showing that every graph with no K_tminor is (2t  6)colorable for t = 7, 8, 9, in the process giving a shorter and computerfree 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 Ktminors is true. Additionally, we show that any graph with no K8?minor is 9colorable, and any graph with no K8?minor is 8colorable. The Kempechain method developed for our proofs of the above results may be of independent interest. We also use Mader's HWege theorem to establish some sufficient conditions for a graph to contain a K8minor.Another motivation for my research is a wellknown conjecture of Erd?s and Lov(&)#225;sz from 1968, the DoubleCritical Graph Conjecture. A connected graph G is doublecritical if for all edges xy ? E(G), ?(G  x  y) = ?(G)  2. Erd?s and Lov(&)#225;sz conjectured that the only doublecritical tchromatic 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 noncomplete, doublecritical, tchromatic graph contains Kt as a minor for t ? 8. We give a shorter proof of this result for t = 7, a computerfree proof for t = 8, and extend the result to show that G contains a K9minor for all t ? 9. Finally, we show that the DoubleCritical Graph Conjecture is true for doublecritical graphs with chromatic number t ? 8 if such graphs are clawfree.
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 Ktminor 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 K7minor is 7colorable. We begin by showing that every graph with no K_tminor is (2t  6)colorable for t = 7, 8, 9, in the process giving a shorter and computerfree 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 Ktminors is true. Additionally, we show that any graph with no K8?minor is 9colorable, and any graph with no K8?minor is 8colorable. The Kempechain method developed for our proofs of the above results may be of independent interest. We also use Mader's HWege theorem to establish some sufficient conditions for a graph to contain a K8minor.Another motivation for my research is a wellknown conjecture of Erd?s and Lov(&)#225;sz from 1968, the DoubleCritical Graph Conjecture. A connected graph G is doublecritical if for all edges xy ? E(G), ?(G  x  y) = ?(G)  2. Erd?s and Lov(&)#225;sz conjectured that the only doublecritical tchromatic 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 noncomplete, doublecritical, tchromatic graph contains Kt as a minor for t ? 8. We give a shorter proof of this result for t = 7, a computerfree proof for t = 8, and extend the result to show that G contains a K9minor for all t ? 9. Finally, we show that the DoubleCritical Graph Conjecture is true for doublecritical graphs with chromatic number t ? 8 if such graphs are clawfree.  
Identifier:  CFE0006649 (IID), ucf:51227 (fedora)  
Note(s): 
20170501 Ph.D. Sciences, Mathematics Doctoral This record was generated from author submitted information. 

Subject(s):  vertex coloring  doublecritical graphs  clawfree graphs  graph minors  Hadwiger's conjecture  
Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0006649  
Restrictions on Access:  public 20170515  
Host Institution:  UCF 