Color Complexity of Recolorable Graph Exploration: Upper and Lower Bounds via Block Structure

Abstract

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 $O(n\Delta)$ moves, where $\Delta$ is the maximum degree.