Boundary Framework, Rear Morphology, and Rectangular Ears in the Partition Graph
Fedor B. Lyudogovskiy

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 , 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 , where is the main chain and are the left and right side edges, and we isolate the nontrivial rectangular family as a canonical discrete family marking the rear part of . We prove that every nontrivial rectangular vertex has degree , has exactly two explicitly described neighbors, and lies in a unique triangle of . This leads to the notions of a rectangular ear, its attachment pair, and its support edge. We also prove that is an…
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