  A b-coloring of a graph is a proper coloring such that each color class has
at least one vertex which is adjacent to each other color class. The b-spectrum
of $G$ is the set $S_{b}(G)$ of integers $k$ such that $G$ has a b-coloring
with $k$ colors and $b(G)=\max S_{b}(G)$ is the b-chromatic number of $G$. A
graph is b-continous if $S_{b}(G)=[\chi(G),b(G)]\cap \mathbb{Z}$. An infinite
number of graphs that are not b-continuous is known. It is also known that
graphs 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 is
adjacent 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 Grundy
coloring.
  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 the
integers 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. These
results generalize previous ones by Linhares-Sales and Silva (2017), and by Shi
et al.(2005).
