Recolorable Graph Exploration by an Oblivious Agent with Fewer Colors

TL;DR

This paper improves the number of colors needed for a recolorable graph exploration algorithm by an oblivious agent, reducing it from eight to six in general and to five for certain restricted graph classes.

Contribution

It introduces a new exploration algorithm that uses fewer colors and proves that five colors suffice for a specific class of graphs, advancing the understanding of recolorable graph exploration.

Findings

Six colors suffice for general graphs.

Five colors suffice for $\

phi\

Abstract

Recently, B\"ockenhauer, 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. B\"ockenhauer~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.