On -improper -coloring of sparse graphs

A graph G is ( 1 , 1 ) -colorable if its vertices can be partitioned into subsets V 1 and V 2 such that every vertex in G [ V i ] has degree at most 1 for each i ∈ { 1 , 2 } . We prove that every graph with maximum average degree at most 14 5 is ( 1 , 1 ) -colorable. In particular, it follows that every planar graph with girth at least 7 is ( 1 , 1 ) -colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to 14 5 (from above) that are not ( 1 , 1 ) -colorable. t, we establish the best possible sufficient condition for the ( 1 , 1 ) -colorability of a graph G in terms of the minimum, ρ G , of ρ G ( S ) = 7 | S | − 5 | E ( G [ S ] ) | over all subsets S of V ( G ) . Namely, every graph G with ρ G ≥ 0 is ( 1 , 1 ) -colorable. On the other hand, we construct infinitely many non- ( 1 , 1 ) -colorable graphs G with ρ G = − 1 . This solves a related conjecture of Kurek and Ruciński from 1994.