A relaxation of the strong Bordeaux Conjecture

Letc 1 , c 2 , … , c kbe k nonnegative integers. A graph G is ( c 1 , c 2 , … , c k ) ‐colorable if the vertex set can be partitioned into k setsV 1 , V 2 , … , V k , such that the subgraph G [ V i ] , induced by V i , has maximum degree at most c i for i = 1 , 2 , … , k . Let F denote the family of plane graphs with neither adjacent 3‐cycles nor 5‐cycles. Borodin and Raspaud (2003) conjectured that each graph in F is (0, 0, 0)‐colorable (which was disproved very recently). In this article, we prove that each graph in F is (1, 1, 0)‐colorable, which improves the results by Xu (2009) and Liu‐Li‐Yu (2016).