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A b-coloring of a graph is a proper coloring such that each color class hasat least one vertex which is adjacent to each other color class. The b-spectrumof $G$ is the set $S_{b}(G)$ of integers $k$ such that $G$ has a b-coloringwith $k$ colors and $b(G)=\max S_{b}(G)$ is the b-chromatic number of $G$. Agraph is b-continous if $S_{b}(G)=[\chi(G),b(G)]\cap \mathbb{Z}$. An infinitenumber of graphs that are not b-continuous is known. It is also known thatgraphs with girth at least 10 are b-continuous.  A partial Grundy coloring is a proper coloring $f:V(G)\rightarrow\{1,\ldots,k\}$ such that each color class $i$ contains some vertex $u$ that isadjacent to every color class $j$ such that $j<i$. The partial Grundy number of$G$ is the maximum value $\partial\Gamma(G)$ for which $G$ has a partial Grundycoloring.  In this work, we prove that graphs with girth at least 8 are b-continuous,and that the b-spectrum of a graph $G$ with girth at least 7 contains theintegers between $2\chi(G)$ and $b(G)$. We also prove that $\partial\Gamma(G)$equals a known upper bound when $G$ is a graph with girth at least 7. Theseresults generalize previous ones by Linhares-Sales and Silva (2017), and by Shiet al.(2005).

b-continuity and Partial Grundy Coloring of graphs with large girth