Graphs and Combinatorics

, Volume 27, Issue 1, pp 73–85

Looseness of Plane Graphs

  • Július Czap
  • Stanislav Jendrol’
  • František Kardoš
  • Jozef Miškuf
Original Paper

DOI: 10.1007/s00373-010-0961-6

Cite this article as:
Czap, J., Jendrol’, S., Kardoš, F. et al. Graphs and Combinatorics (2011) 27: 73. doi:10.1007/s00373-010-0961-6

Abstract

A face of a vertex coloured plane graph is called loose if the number of colours used on its vertices is at least three. The looseness of a plane graph G is the minimum k such that any surjective k-colouring involves a loose face. In this paper we prove that the looseness of a connected plane graph G equals the maximum number of vertex disjoint cycles in the dual graph G* increased by 2. We also show upper bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graphs. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained.

Keywords

Vertex colouring Loose colouring Looseness Plane graph Dual graph 

Mathematics Subject Classification (2000)

05C10 05C15 05C38 

Copyright information

© Springer 2010

Authors and Affiliations

  • Július Czap
    • 1
  • Stanislav Jendrol’
    • 1
  • František Kardoš
    • 1
  • Jozef Miškuf
    • 1
  1. 1.Institute of MathematicsP. J. Šafárik UniversityKošiceSlovakia