  In a bounded max-coloring of a vertex/edge weighted graph, each color class
is of cardinality at most $b$ and of weight equal to the weight of the heaviest
vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask
for such a coloring minimizing the sum of all color classes' weights.
  In this paper we present complexity results and approximation algorithms for
those problems on general graphs, bipartite graphs and trees. We first show
that both problems are polynomial for trees, when the number of colors is
fixed, and $H_b$ approximable for general graphs, when the bound $b$ is fixed.
For the bounded max-vertex-coloring problem, we show a 17/11-approximation
algorithm for bipartite graphs, a PTAS for trees as well as for bipartite
graphs when $b$ is fixed. For unit weights, we show that the known 4/3 lower
bound for bipartite graphs is tight by providing a simple 4/3 approximation
algorithm. For the bounded max-edge-coloring problem, we prove approximation
factors of $3-2/\sqrt{2b}$, for general graphs, $\min\{e, 3-2/\sqrt{b}\}$, for
bipartite graphs, and 2, for trees. Furthermore, we show that this problem is
NP-complete even for trees. This is the first complexity result for
max-coloring problems on trees.
