Fault-Tolerant Designs of Graphs with Gallai’s Property in Euclidean Space Tilings

This study examines graphs that demonstrate Gallai’s property, particularly those in which for every prescribed set S of vertices with $\left|S\right|=j$ there exists a longest path or cycle that avoids that set. Such graphs are naturally fault-tolerant in the structural sense: if some vertices fail, there can still exist longest routes that bypass the failed vertices. Our main purpose is to construct explicit Gallai-type graphs that admit embeddings into a rigorously defined three-dimensional geometric adjacency structure derived from an icosahedral–tetrahedral polyhedral cell complex. We show that similar graphs may be found in three-dimensional structures obtained from a periodic polyhedral packing (cell complex) built from tetrahedral and icosahedral cells. Importantly, we do not claim a face-to-face tessellation of ${\mathbb{R}}^3$ by congruent regular icosahedra and tetrahedra; instead, we define a specific periodic cell complex ${\mathcal{IT}}^3$ and work in its associated adjacency graph $\mathsf{\Gamma }\left({\mathcal{IT}}^3\right)$. These geometric constructions expand lattice-based findings to a three-dimensional adjacency setting and provide new embeddings for Gallai-type graphs. Connections to AI systems are mentioned at the conceptual level.