Article

Looseness of Plane Graphs

Graphs and Combinatorics (impact factor: 0.32). 04/2012; 27(1):73-85. DOI:10.1007/s00373-010-0961-6 pp.73-85
Source: DBLP

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.

KeywordsVertex colouring-Loose colouring-Looseness-Plane graph-Dual graph
Mathematics Subject Classification (2000)05C10-05C15-05C38

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Keywords

connected plane graph G
 
dual graph G*
 
dual graphs
 
edge connectivity
 
girth
 
graphs
 
loose
 
loose face
 
looseness
 
maximum number
 
minimum k
 
Negami
 
plane graph G
 
plane triangulations
 
present infinite classes
 
surjective k-colouring
 
vertex coloured plane graph
 
vertex disjoint cycles