From the plane to higher surfaces

  • Authors:

  • Ken-Ichi Kawarabayashi;Carsten Thomassen

  • Affiliations:

  • National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan;Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark and King Abdulaziz University, Jeddah, Saudi Arabia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

Quantified Score

Hi-index 0.00

Visualization

Abstract

We show that Grotzsch@?s theorem extends to all higher surfaces in the sense that every triangle-free graph on a surface of Euler genus g becomes 3-colorable after deleting a set of at most 1000@?g@?f(g) vertices where f(g) is the smallest edge-width which guarantees a graph of Euler genus g and girth 5 to be 3-colorable. We derive this result from a general cutting technique which we also use to extend other results on planar graphs to higher surfaces in the same spirit, even after deleting only 1000g vertices. These include the 5-list-color theorem, results on arboricity, and various types of colorings, and decomposition theorems of planar graphs into two graphs with prescribed degeneracy properties. It is not known if the 4-color theorem extends in this way.