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.

翻译：我们研究了一种约束性极强的图探索问题。在该模型中，一个无持久记忆的智能体被放置在图的一个顶点上，仅能观测到相邻顶点。目标是访问图中的每个顶点，返回起始顶点并终止。智能体不知道通过哪条边进入当前顶点。智能体可为当前顶点着色，并能以任意顺序观测相邻顶点的颜色。智能体不得对顶点重新着色。我们研究了探索所有图所需的最少颜色数量及其充分性。证明了一般情况下n-1种颜色对图探索是必要且充分的；仅探索树时，3种颜色是必要且充分的；对于大小为n、周长为k（周长定义为最长环的长度）的图，所需颜色数的下界为min(2k-3,n-1)，上界为min(2k-1,n-1)。这一结论仅适用于需要探索所有图类而非特定图类的算法。我们给出了一个图类的示例，其中每个图均可用4种颜色探索，尽管这些图具有最大周长。此外，我们证明了顶点重新着色具有强大的能力：通过设计一个允许重新着色的算法，仅需7种颜色即可探索所有图。