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Recolorable Graph Exploration by an Oblivious Agent with Fewer Colors

Shota Takahashi, Haruki Kanaya, Shoma Hiraoka, Ryota Eguchi, Yuichi Sudo

TL;DR

The paper advances recolorable graph exploration by an oblivious agent on unknown, port-less graphs, reducing color complexity from eight to six for arbitrary graphs and to five for triangle-free and $\varphi$-free graphs. It achieves this via a DFS-inspired semi-DFS framework and a fixed-color simulation ($\mathcal{A}_{Gen6}$) that propagates a path and handles backtracking with a limited color set. A separate $\mathcal{A}_{PF5}$ demonstrates five-color exploration on $\varphi$-free graphs by enforcing an acyclic distance-color DAG, ensuring reachability and termination. The work also includes a model-clarification discussion about recoloring rules and color counting, clarifying assumptions and outlining open questions such as potential further color reductions or memory-enabled variants.

Abstract

Recently, Böckenhauer, Frei, Unger, and Wehner (SIROCCO 2023) introduced a novel variant of the graph exploration problem in which a single memoryless agent must visit all nodes of an unknown, undirected, and connected graph before returning to its starting node. Unlike the standard model for mobile agents, edges are not labeled with port numbers. Instead, the agent can color its current node and observe the color of each neighboring node. To move, it specifies a target color and then moves to an adversarially chosen neighbor of that color. Böckenhauer~et al.~analyzed the minimum number of colors required for successful exploration and proposed an elegant algorithm that enables the agent to explore an arbitrary graph using only eight colors. In this paper, we present a novel graph exploration algorithm that requires only six colors. Furthermore, we prove that five colors are sufficient if we consider only a restricted class of graphs, which we call the $\varphi$-free graphs, a class that includes every graph with maximum degree at most three and every cactus.

Recolorable Graph Exploration by an Oblivious Agent with Fewer Colors

TL;DR

The paper advances recolorable graph exploration by an oblivious agent on unknown, port-less graphs, reducing color complexity from eight to six for arbitrary graphs and to five for triangle-free and -free graphs. It achieves this via a DFS-inspired semi-DFS framework and a fixed-color simulation () that propagates a path and handles backtracking with a limited color set. A separate demonstrates five-color exploration on -free graphs by enforcing an acyclic distance-color DAG, ensuring reachability and termination. The work also includes a model-clarification discussion about recoloring rules and color counting, clarifying assumptions and outlining open questions such as potential further color reductions or memory-enabled variants.

Abstract

Recently, Böckenhauer, Frei, Unger, and Wehner (SIROCCO 2023) introduced a novel variant of the graph exploration problem in which a single memoryless agent must visit all nodes of an unknown, undirected, and connected graph before returning to its starting node. Unlike the standard model for mobile agents, edges are not labeled with port numbers. Instead, the agent can color its current node and observe the color of each neighboring node. To move, it specifies a target color and then moves to an adversarially chosen neighbor of that color. Böckenhauer~et al.~analyzed the minimum number of colors required for successful exploration and proposed an elegant algorithm that enables the agent to explore an arbitrary graph using only eight colors. In this paper, we present a novel graph exploration algorithm that requires only six colors. Furthermore, we prove that five colors are sufficient if we consider only a restricted class of graphs, which we call the -free graphs, a class that includes every graph with maximum degree at most three and every cactus.
Paper Structure (16 sections, 8 theorems, 11 equations, 5 figures, 4 algorithms)

This paper contains 16 sections, 8 theorems, 11 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Preliminaries
  4. Exploration on Arbitrary Graphs
  5. Semi-DFS
  6. Termination:
  7. Full Visitation:
  8. How to simulate Semi-DFS with Six Colors
  9. Case 1. $U(P,F)\neq\emptyset$:
  10. Case 2. $U(P,F)=\emptyset$:
  11. Triangle-free Case
  12. Exploration on $\varphi$-free Graphs
  13. Model Clarification
  14. The Omitted Proof for Triangle-free Case
  15. Case 1. $U(P,F) \neq \emptyset$
  16. ...and 1 more sections

Key Result

Lemma 1

For any graph $G=(V,E)$ and for any starting node $s\in V$, the semi-DFS algorithm eventually terminates, and by the time it terminates, every node in $G$ has been visited (i.e., $F=V$ eventually holds).

Figures (5)

  • Figure 1: Lifecycle of node colors in algorithm $\mathcal{A}_{\textsc{Gen6}}$
  • Figure 2: Path expansion
  • Figure 3: Removal of the head
  • Figure 4: $\varphi$-shaped graphs
  • Figure 5: $\varphi$-shaped graph in the proof of Lemma \ref{['lemma:phi-free']}

Theorems & Definitions (18)

  • Definition 1: Graph Exploration
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma:semi-dfs']}
  • Theorem 1
  • proof
  • Theorem 2
  • Definition 2: $\varphi$-free graph
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 8 more