List r-hued chromatic number of graphs with bounded maximum average degrees

For integers k,r>0, a (k,r)-coloring of a graph G is a proper coloring c with at most k colors such that for any vertex v with degree d(v), there are at least min{d(v),r} different colors present at the neighborhood of v. The r-hued chromatic number of G, χr(G), is the least integer k such that a (k,r)-coloring of G exists. The listr-hued chromatic numberχL,r(G) of G is similarly defined. Thus if Δ(G)≥r, then χL,r(G)≥χr(G)≥r+1. We present examples to show that, for any sufficiently large integer r, there exist graphs with maximum average degree less than 3 that cannot be (r+1,r)-colored. We prove that, for any fraction q<145, there exists an integer R=R(q) such that for each r≥R, every graph G with maximum average degree q is list (r+1,r)-colorable. We present examples to show that for some r there exist graphs with maximum average degree less than 4 that cannot be r-hued colored with less than 3r2 colors. We prove that, for any sufficiently small real number ϵ>0, there exists an integer h=h(ϵ) such that every graph G with maximum average degree 4−ϵ satisfies χL,r(G)≤r+h(ϵ). These results extend former results in Bonamy et al. (2014).