Zero-Memory Graph Exploration with Unknown Inports

TL;DR

This paper investigates the minimal number of vertex colors needed for a memoryless agent to explore all graphs, revealing tight bounds for various graph classes and demonstrating the power of recoloring with a small number of colors.

Contribution

It establishes tight bounds on the number of colors required for graph exploration by a memoryless agent and shows how recoloring significantly reduces this requirement.

Findings

n-1 colors are necessary and sufficient for general graph exploration

3 colors suffice for exploring trees

Recoloring reduces the number of colors needed to 7 for all graphs

Abstract

We study a very restrictive graph exploration problem. In our model, an agent without persistent memory is placed on a vertex of a graph and only sees the adjacent vertices. The goal is to visit every vertex of the graph, return to the start vertex, and terminate. The agent does not know through which edge it entered a vertex. The agent may color the current vertex and can see the colors of the neighboring vertices in an arbitrary order. The agent may not recolor a vertex. We investigate the number of colors necessary and sufficient to explore all graphs. We prove that n-1 colors are necessary and sufficient for exploration in general, 3 colors are necessary and sufficient if only trees are to be explored, and min(2k-3,n-1) colors are necessary and min(2k-1,n-1) colors are sufficient on graphs of size n and circumference $k$, where the circumference is the length of a longest cycle. This only holds if an algorithm has to explore all graphs and not merely certain graph classes. We give an example for a graph class where each graph can be explored with 4 colors, although the graphs have maximal circumference. Moreover, we prove that recoloring vertices is very powerful by designing an algorithm with recoloring that uses only 7 colors and explores all graphs.