We study the problem of online graph coloring for $k$-colorable graphs. The best previously known deterministic algorithm uses $\widetilde{O}(n^{1-\frac{1}{k!}})$ colors for general $k$ and $\widetilde{O}(n^{5/6})$ colors for $k = 4$, both given by Kierstead in 1998. In this paper, we finally break this barrier, achieving the first major improvement in nearly three decades. Our results are summarized as follows: (1) $k \geq 5$ case. We provide a deterministic online algorithm to color $k$-colorable graphs with $\widetilde{O}(n^{1-\frac{1}{k(k-1)/2}})$ colors, significantly improving the current upper bound of $\widetilde{O}(n^{1-\frac{1}{k!}})$ colors. Our algorithm also matches the best-known bound for $k = 4$ ($\widetilde{O}(n^{5/6})$ colors). (2) $k = 4$ case. We provide a deterministic online algorithm to color $4$-colorable graphs with $\widetilde{O}(n^{14/17})$ colors, improving the current upper bound of $\widetilde{O}(n^{5/6})$ 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 a factor of $1.09$. 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 $O(\frac{n}{\log \log n})$, which improves the best-known result of $O(\frac{n \log \log \log n}{\log \log n})$ 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.

翻译：我们研究可$k$染色图的在线图着色问题。此前已知最优的确定性算法对于一般$k$使用$\widetilde{O}(n^{1-\frac{1}{k!}})$种颜色，对于$k=4$使用$\widetilde{O}(n^{5/6})$种颜色，二者均由Kierstead于1998年提出。本文最终突破这一屏障，实现了近三十年来的首次重大改进。我们的结果总结如下：(1) $k \geq 5$情形：我们提出一种确定性在线算法，能以$\widetilde{O}(n^{1-\frac{1}{k(k-1)/2}})$种颜色对可$k$染色图着色，显著改进了目前$\widetilde{O}(n^{1-\frac{1}{k!}})$种颜色的上界。该算法在$k=4$情形下也匹配已知最优界（$\widetilde{O}(n^{5/6})$种颜色）。(2) $k=4$情形：我们提出一种确定性在线算法，能以$\widetilde{O}(n^{14/17})$种颜色对可4染色图着色，改进了目前$\widetilde{O}(n^{5/6})$种颜色的上界。(3) $k=2$情形：我们证明随机算法的上界为$1.034 \log_2 n + O(1)$种颜色，下界为$\frac{91}{96} \log_2 n - O(1)$种颜色，这表明我们将间隙缩小至因子$1.09$。借助$k \geq 5$情形的算法，我们还获得一种确定性在线图着色算法，其竞争比为$O(\frac{n}{\log \log n})$，改进了Kierstead给出的最佳已知结果$O(\frac{n \log \log \log n}{\log \log n})$。对于二分图情形（$k=2$），在线确定性算法的极限已知：任何确定性算法都需要$2 \log_2 n - O(1)$种颜色。我们的结果意味着随机算法虽能略优但仍存在极限。