  The chromatic number of a graph is the minimum $k$ such that the graph has a
proper $k$-coloring. There are many coloring parameters in the literature that
are proper colorings that also forbid bicolored subgraphs. Some examples are
$2$-distance coloring, acyclic coloring, and star coloring, which forbid a
bicolored path on three vertices, bicolored cycles, and a bicolored path on
four vertices, respectively. This notion was first suggested by Gr\"unbaum in
1973, but no specific name was given. We revive this notion by defining an
$H$-avoiding $k$-coloring to be a proper $k$-coloring that forbids a bicolored
subgraph $H$.
  When considering the class $\mathcal C$ of graphs with no $F$ as an induced
subgraph, it is not hard to see that every graph in $\mathcal C$ has bounded
chromatic number if and only if $F$ is a complete graph of size at most two. We
study this phenomena for the class of graphs with no $F$ as a subgraph for
$H$-avoiding coloring. We completely characterize all graphs $F$ where the
class of graphs with no $F$ as a subgraph has bounded $H$-avoiding chromatic
number for a large class of graphs $H$. As a corollary, our main result implies
a characterization of graphs $F$ where the class of graphs with no $F$ as a
subgraph has bounded star chromatic number. We also obtain a complete
characterization for the acyclic chromatic number.
