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Arrangeability and Clique Subdivisions

  • Vojtěch RödlEmail author
  • Robin Thomas
Chapter

Summary

Let k be an integer. A graph G is k-arrangeable (concept introduced by Chen and Schelp) if the vertices of G can be numbered v 1, v 2, , v n in such a way that for every integer i with 1 ≤ in, at most k vertices among {v 1, v 2, , v i } have a neighbor \(v \in \{ v_{i+1},v_{i+2},\ldots,v_{n}\}\) that is adjacent to v i . We prove that for every integer p ≥ 1, if a graph G is not 2500(p + 1)8-arrangeable, then it contains a K p -subdivision. By a result of Chen and Schelp this implies that graphs with no K p -subdivision have “linearly bounded Ramsey numbers,” and by a result of Kierstead and Trotter it implies that such graphs have bounded “game chromatic number.”

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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