Recolorable Graph Exploration by an Oblivious Agent with Fewer Colors

Recolorable Graph Exploration by an Oblivious Agent with Fewer Colors (arXiv:2505.02789v2) is a paper by Shota Takahashi, Haruki Kanaya, Shoma Hiraoka, Ryota Eguchi, and Yuichi Sudo that reduces the color complexity for exploring unknown, port-free graphs using a memoryless agent.

• Arbitrary Graphs: The authors present a new algorithm requiring only six colors, improving upon the previous eight-color bound established by Böckenhauer et al. This is achieved by simulating a semi-DFS traversal that maintains a chord-free return path.
• Triangle-Free Graphs: For graphs containing no cycles of length three, the algorithm can be adapted to use just five colors.
• $\phi$-Free Graphs: The paper proves that five colors suffice for $\phi$-free graphs, a class that includes all graphs with a maximum degree of at most three and all cactus graphs.

The work addresses an ambiguity in the original model regarding whether nodes can be recolored back to their initial state; under the interpretation allowing this, the six-color bound holds, whereas forbidding it would raise the arbitrary graph bound to seven. The paper was accepted to OPODIS 2025 (Leibniz International Proceedings in Informatics).

The paper studies a constrained version of unknown-graph exploration in which a single deterministic, memoryless agent must visit every node of an unknown, connected, undirected graph and then return to its starting node. The setting departs from the standard mobile-agent model by removing port numbers: incident edges are not locally labeled, so the agent cannot rely on fixed edge identifiers to remember or distinguish where it has been. Instead, the agent operates in a recolorable environment, using a finite palette of colors to mark or reconfigure edges as it moves. The core question is how little environmental state—measured by the number of colors available—is sufficient for an oblivious agent to perform complete exploration and return.

The main contribution is an improved recoloring strategy that enables such an agent to explore arbitrary connected graphs with fewer colors than prior constructions, including the related model introduced by Böckenhauer, Frei, Unger, and Wehner. Rather than treating colors as simple visited/unvisited flags, the algorithm uses them to encode enough local information for a memoryless controller to maintain a coherent search process, recognize already processed structure, and eventually unwind back to the start. The work thus advances the understanding of the “color complexity” of oblivious graph exploration: how much environmental marking power is needed when the agent itself has no memory and the graph provides no port-number labels.

This matters because memoryless agents are attractive in low-power or highly constrained settings, such as robotic probes, network crawlers, or physical systems where only simple markings can be persisted. By showing that exploration and return can be achieved with a smaller color palette, the paper demonstrates that environmental memory can be used more economically, narrowing the gap between idealized mobile-agent models and practical constraints. It also clarifies what remains genuinely difficult when edge identifiers are absent and only limited recoloring is available.