  We introduce a new type of adversary for online graph problems. The new
adversary is parameterized by a single integer $\kappa$, which upper bounds the
number of connected components that the adversary can use at any time during
the presentation of the online graph $G$. We call this adversary "$\kappa$
components bounded", or $\kappa$-CB for short. On one hand, this adversary is
restricted compared to the classical adversary because of the $\kappa$-CB
constraint. On the other hand, we seek competitive ratios parameterized only by
$\kappa$ with no dependence on the input length $n$, thereby giving the new
adversary power to use arbitrarily large inputs.
  We study online coloring under the $\kappa$-CB adversary. We obtain finer
analysis of the existing algorithms $FirstFit$ and $CBIP$ by computing their
competitive ratios on trees and bipartite graphs under the new adversary.
Surprisingly, $FirstFit$ outperforms $CBIP$ on trees. When it comes to
bipartite graphs $FirstFit$ is no longer competitive under the new adversary,
while $CBIP$ uses at most $2\kappa$ colors. We also study several well known
classes of graphs, such as $3$-colorable, $C_k$-free, $d$-inductive, planar,
and bounded treewidth, with respect to online coloring under the $\kappa$-CB
adversary. We demonstrate that the extra adversarial power of unbounded input
length outweighs the restriction on the number of connected components leading
to non existence of competitive algorithms for these classes.
