发表机构
The University of Osaka; Hosei University（大阪大学; 法政大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无记忆智能体探索匿名图所需的最小颜色数，通过块结构给出上界与下界，证明树和环需三色，块为环或完全二分图的图需四色，并改进仙人掌图的上界至四色。
AI 中文摘要
我们研究单个无内部记忆的智能体对匿名、无端口图的探索。为弥补记忆的缺失，智能体使用可写的顶点颜色作为外部记忆。从每个起始顶点出发，智能体必须访问所有顶点，返回起点，并在那里终止。在整个过程中，重着色不受限制，颜色数量包括共同的初始颜色。然而，据我们所知，对于不受限制的重着色，此前没有已知的非平凡颜色下界。我们在由块结构定义的两个类别上确定了最优颜色数量，并证明了不受限制重着色的首个非平凡颜色下界。首先，一个三色算法以$O(n)$步探索每棵树和每个简单环，且任何至多两色的算法无法探索$P_3$（三个顶点的路径）。其次，我们给出一个四色算法，以$O(n)$步探索每个块为环或完全二分图的图，并证明任何至多三色的算法无法探索所有次立方伪树。因此，对于次立方伪树与该块定义类别之间的每个类别，四色是最优的。对于仙人掌图，这改进了先前的五色上界，使之紧致为四色。下界通过手工减少可能的初始动作，并通过一个机器检查的SAT证书排除了九个至多五个顶点的图上的其余情况。最后，我们将已知的三角形自由图五色算法扩展到块为团或三角形自由的图，使用$O(n\Delta)$步，其中$\Delta$是最大度。
英文摘要
We study exploration of anonymous, port-free graphs by a single agent with no internal memory. To compensate for the lack of memory, the agent uses writable vertex colors as external memory. From every starting vertex, the agent must visit all vertices, return to its start, and terminate there. Throughout, recoloring is unrestricted, and the color count includes the common initial color. However, to our knowledge, no nontrivial color lower bound was known for unrestricted recoloring. We determine the optimal number of colors on two classes defined by block structure and prove the first nontrivial color lower bounds for unrestricted recoloring. First, a single three-color algorithm explores every tree and every simple cycle in $O(n)$ moves, and no algorithm with at most two colors explores $P_3$, the path on three vertices. Second, we give a four-color algorithm that explores every graph whose blocks are cycles or complete bipartite graphs in $O(n)$ moves, and we prove that no algorithm with at most three colors explores all subcubic pseudotrees. Hence four colors are optimal for every class between subcubic pseudotrees and this block-defined class. On cacti, this improves the previous five-color upper bound to a tight four. The lower bound reduces the possible initial actions by hand and rules out the remaining cases by a machine-checked SAT certificate on nine graphs with at most five vertices. Finally, we extend the known five-color algorithm for triangle-free graphs to graphs whose blocks are cliques or triangle-free, using $O(nΔ)$ moves, where $Δ$ is the maximum degree.
Comments23 pages, 4 figures, 2 tables
