Online Graph Coloring for $k$-Colorable Graphs

We study the problem of online graph coloring for $k$-colorable graphs. The best previously known deterministic algorithm uses $\tilde{O}(n^{1-1/k!})$ colors for general $k$ and $\tilde{O}(n^{5/6})$ colors for $k = 4$, both given by Kierstead in 1998. In this paper, nearly thirty years later, we have finally made progress. Our results are summarized as follows: (1) $k \geq 5$ case. We provide a deterministic online algorithm to color $k$-colorable graphs with $\tilde{O}(n^{1-2/(k(k-1))})$ colors, significantly improving the current upper bound of $\tilde{O}(n^{1-1/k!})$ (2) $k = 4$ case. We provide a deterministic online algorithm to color $4$-colorable graphs with $\tilde{O}(n^{14/17})$ colors, improving the current upper bound of $\tilde{O}(n^{5/6})$ colors. (3) $k = 2$ case. We show that for randomized algorithms, the upper bound is $1.034 \log_2 n + O(1)$ colors and the lower bound is $\frac{91}{96} \log_2 n - O(1)$ colors. This means that we close the gap to $1.09\mathrm{x}$. With our algorithm for the $k \geq 5$ case, we also obtain a deterministic online algorithm for graph coloring that achieves a competitive ratio of $O(n / \log \log n)$, which improves the best known result of $O(n \log \log \log n / \log \log n)$ by Kierstead. For the bipartite graph case ($k = 2$), the limit of online deterministic algorithms is known: any deterministic algorithm requires $2 \log_2 n - O(1)$ colors. Our results imply that randomized algorithms can perform slightly better but still have a limit.