Matchings and Hadwiger's conjecture

  • Authors:

  • Andrei Kotlov

  • Affiliations:

  • CWI, Kruislaan 413, 1098 SJ Amsterdam, Netherlands

  • Venue:
  • Discrete Mathematics - Algebraic and topological methods in graph theory
  • Year:
  • 2002

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Abstract

Assuming that a graph G on n vertices is a minimal counterexample to Hadwiger's Conjecture χ(G) ≤ η(G), we apply the Edmonds-Gallai Structure Theorem to its complement, H, to find that H has a matching of size ⌊n/2⌋. Hence Magyar Tud. Acad. Mat. Kutató Int. Közl. 8 (1963) 373: χ(G) ≤ ⌈n/2⌉. Further, H is homeomorphic to a three-connected graph, and is of tree width at least four. The same holds for a minimal counterexample G to Colin de Verdière's Conjecture µ(G) + 1 ≥ χ(G).