Path partition of planar graphs with girth at least six.

A graph G has a ( k 1 , k 2 ) -partition if V (G) can be partitioned into two nonempty disjoint subsets V 1 and V 2 so that G [ V 1 ] and G [ V 2 ] are graphs whose components are paths of order at most k 1 and k 2 , respectively. In this paper, we proved that every planar graph with girth at least six giving that i -cycle is not intersecting with j -cycle admits a ( 3 , 3) -partition, where i ∈ { 6 , 7 } and j ∈ { 6 , 7 , 8 , 9 }. [ABSTRACT FROM AUTHOR]