Boundary Framework, Rear Morphology, and Rectangular Ears in the Partition Graph

Abstract

We study the outer geometry of the partition graph $G_n$, focusing on its canonical front-and-side framework, the family of nontrivial rectangular partitions, and the rear structures suggested by the visible geometry of the graph. We formalize the boundary framework $\mathcal B_n=\mathcal M_n\cup\mathcal L_n\cup\mathcal R_n$, where $\mathcal M_n$ is the main chain and $\mathcal L_n,\mathcal R_n$ are the left and right side edges, and we isolate the nontrivial rectangular family $\mathrm{Rect}^*(n)=\{(a^b):ab=n,\ a,b\ge2\}$ as a canonical discrete family marking the rear part of $G_n$. We prove that every nontrivial rectangular vertex $\rho=(a^b)$ has degree $2$, has exactly two explicitly described neighbors, and lies in a unique triangle of $G_n$. This leads to the notions of a rectangular ear, its attachment pair, and its support edge. We also prove that $\mathrm{Rect}^*(n)$ is an independent set in $G_n$, so the weak rectangular contour is not a graph-theoretic chain but a discrete rear marker family. For every genuinely rear rectangular ear, namely for $a,b\ge3$, we show that its support edge lies in a tetrahedral configuration of the clique complex $K_n=\mathrm{Cl}(G_n)$. To organize the interaction between different ears, we introduce support zones, support distances, and support corridors between attachment pairs. The paper also records a natural divisor-theoretic indexing of the rectangular family, presents a computational atlas in small and large ranges, and concludes with open problems concerning support-zone connectivity, inter-ear corridors, and canonical rear contours in $G_n$.