In a proper vertex coloring c of a graph G, a vertex u is called a b-vertex if u is adjacent to a vertex in every other color class. A b∗-coloring is a proper coloring in which a b-vertex is adjacent to a b-vertex in every other color class. A Grundy coloring is a proper coloring obtained by the First-Fit (greedy) coloring procedure. A
Nearby in the stack
z
-coloring of
G
is a
b∗
-coloring that is also a Grundy coloring. The
b∗
-chromatic number (resp.,
z
-chromatic number), denoted by
b∗(G)
(resp.,
z(G)
), is the maximum number of colors used in a
b∗
-coloring (resp.,
z
-coloring) of
G
. Every graph admits a
b∗
-coloring and a
z
-coloring that can be found using a polynomial-time coloring heuristic. Let
m∗(G)
be the largest integer
k
such that a vertex of degree at least
k
in
G
has
k
neighbors of degree at least
k
. We employ list-coloring techniques to prove that if
G
has a girth of at least
7
, then
b∗(G)=m∗(G)+1
. A similar result is obtained for graphs of girth at least
6
when
m∗=3
. Finally, we obtain some results for the
z
-chromatic number. We prove that if the girth is at least
2m∗(G)+4
and
G
contains a specific tree as an ordinary subgraph, then
z(G)=m∗(G)+1
.
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