From χ- to χ-bounded classes

χ-bounded classes are studied here in the context of star colorings and, more generally, χp-colorings. This fits to a general scheme sparsity and leads natural extensions notion bounded expansion class. In this paper we solve two conjectures related coloring (i.e. χ2) boundedness. One is disproved fact determine which weakening holds true. χp-boundedness stability give structural characterizations (strong weak) χp-bounded classes. We also generalize result Wood relating chromatic number graph its 1-subdivision. As an application our characterizations, among other things, show that for every odd integer g>3 even hole-free graphs G contain at most φ(g,ω(G))|G| holes length g.