Online Graph Coloring for $k$-Colorable Graphs

2025-11-20

AI Summary

This paper studies the online graph coloring problem for $k$-colorable graphs, aiming to reduce the upper bound on the number of colors used and improve the competitive ratio. For $k geq 5$, $k = 4$, and $k = 2$, we propose novel deterministic and randomized online algorithms—marking the first improvement over Kierstead’s long-standing upper bound in nearly three decades. Our analysis yields: (i) an upper bound of $ ilde{O}(n^{1 - 2/(k(k-1))})$ for $k geq 5$; (ii) $ ilde{O}(n^{14/17})$ for $k = 4$; and (iii) a lower bound of approximately $1.034 log_2 n$ for the randomized case with $k = 2$. Methodologically, we integrate deterministic framework design, probabilistic strategy analysis, and refined asymptotic complexity estimation. These advances significantly narrow the gap between known theoretical upper and lower bounds for online graph coloring, thereby pushing forward the state-of-the-art in competitive ratio performance for this classical problem.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the problem of online graph coloring for $k$-colorable graphs. The best previously known deterministic algorithm uses $ ilde{O}(n^{1-1/k!})$ colors for general $k$ and $ ilde{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 $ ilde{O}(n^{1-2/(k(k-1))})$ colors, significantly improving the current upper bound of $ ilde{O}(n^{1-1/k!})$ (2) $k = 4$ case. We provide a deterministic online algorithm to color $4$-colorable graphs with $ ilde{O}(n^{14/17})$ colors, improving the current upper bound of $ ilde{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 $1.09mathrm{x}$. 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.

Problem

• Developing deterministic online algorithms for k-colorable graphs
• Improving color bounds for k=4 and k≥5 cases significantly
• Analyzing randomized algorithm limits for bipartite graph coloring

Innovation

• Deterministic online coloring for k-colorable graphs
• Improved upper bounds for k=4 and k≥5 cases
• Randomized algorithm with logarithmic bound for k=2