Planar graphs without adjacent cycles of length at most five are (1,1,0) -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 i = 1 , 2 , … , k . Borodin, Montassier and Raspaud asked: Is every planar graph without adjacent cycles of length at most five 3-colorable, i.e., ( 0 , 0 , 0 ) -colorable? This problem has now been answered negatively by Cohen-Addad et al. who successfully constructed a non-3-colorable planar graph with neither 4-cycles nor 5-cycles. Is every planar graph without adjacent cycles of length at most five ( 1 , 0 , 0 ) -colorable? To this new problem, this paper proves that every planar graph without adjacent cycles of length at most five is ( 1 , 1 , 0 ) -colorable. [ABSTRACT FROM AUTHOR]