χ-bounded classes are studied here in the context of star colorings and, more generally, ${\chi }_p$-colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. ${\chi }_2$) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. ${\chi }_p$-boundedness leads to more stability and we give structural characterizations of (strong and weak) ${\chi }_p$-bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer $g>3$ even hole-free graphs G contain at most $\phi (g,\omega (G))|G|$ holes of length g.

χ 有界类在此在星形着色的背景下进行研究，更一般地说，${\chi }_{磷}$-着色。这符合稀疏性的一般方案，并导致有界扩展类概念的自然扩展。在本文中，我们解决了与星星着色相关的两个猜想（即${\chi }_2$) 有界。其中一个猜想被推翻，实际上我们确定了哪种弱化是正确的。${\chi }_{磷}$- 有界导致更多的稳定性，我们给出了（强和弱）的结构特征 ${\chi }_{磷}$-有界类。我们还概括了 Wood 将图的色数与其 1 细分的星色数相关联的结果。作为我们特征的应用，除其他外，我们证明对于每个奇数整数$G>3$甚至无孔图G最多包含$\phi (G,\omega (G))|G|$长度为g 的孔。