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

TL;DR

This paper analyzes the boundary and rear structures of the partition graph, introducing a boundary framework, rectangular families, and geometric configurations, with implications for understanding the graph's topology and connectivity.

Contribution

It formalizes the boundary framework and rectangular family in the partition graph, revealing their structural properties and introducing new concepts like rectangular ears and support zones.

Findings

Rectangular vertices have degree 2 and are part of a unique triangle.

The rectangular family forms an independent set, serving as a rear marker.

Support edges of rear rectangular ears lie in tetrahedral configurations.

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 $G_n$.