Planar graphs with cycles of length neither 4 nor 7 are -colorable

Let d 1 , d 2 , … , d k be k non-negative integers. A graph G is ( d 1 , d 2 , … , d k ) -colorable if the vertex set of G can be partitioned into subsets V 1 , V 2 , … , V k such that the subgraph G [ V i ] induced by V i has maximum degree at most d i for 1 ≤ i ≤ k . It is known that planar graphs with cycles of length neither 4 nor k , k ∈ { 5 , 6 } , are ( 3 , 0 , 0 ) -colorable. In this paper, we show that planar graphs with cycles of length neither 4 nor 7 are also ( 3 , 0 , 0 ) -colorable.