Online Coloring and a New Type of Adversary for Online Graph Problems

We introduce a new type of adversary for online graph problems. The new adversary is parameterized by a single integer $κ$, 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 "$κ$ components bounded", or $κ$-CB for short. On one hand, this adversary is restricted compared to the classical adversary because of the $κ$-CB constraint. On the other hand, we seek competitive ratios parameterized only by $κ$ 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 $κ$-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 $2κ$ 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 $κ$-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.