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arXiv · 2606.20773

Local Complexity, Overlap Control, and Global Simplicity in Partition Graphs: A Conceptual Synthesis for G_n and K_n = Cl(G_n)

Fedor B. Lyudogovskiy
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Abstract

For each positive integer $n$, let $G_n$ be the partition graph whose vertices are the partitions of $n$, with adjacency defined by an elementary transfer of one unit between parts, followed by reordering. Let $K_n=\mathrm{Cl}(G_n)$ be its clique complex. Previous work revealed two apparently contrasting features of this family. Locally, the graphs $G_n$ become increasingly rich: local clique dimensions grow, degree landscapes refine, support jumps and simplex layers proliferate, and axial, rear, central, shell, and directional structures become more pronounced. Globally, however, the clique complexes remain homotopically simple: $K_n\simeq \bigvee^{b_n}S^2$, where $b_n=χ(K_n)-1$. This paper gives a conceptual synthesis explaining why these facts are compatible. The local side is governed by ordered local transfer types, which determine local neighborhood graphs, degrees, local clique numbers, and local simplex dimensions. The global side is governed not by the largest local simplices, but by the overlap pattern of canonical full star/top simplices. These simplices form a good cover $\mathcal C_n$, giving $K_n\simeq N(\mathcal C_n)=N_n$. The nerve admits an anchor-cover and intersection-poset reduction $N_n\simeq Δ(J_n)$, with $\dimΔ(J_n)\le 2$. Thus high-dimensional local simplices occur inside contractible containers, while global topology is controlled by a low-dimensional overlap poset. Within this family, the qualitative topological problem therefore collapses to the numerical computation and interpretation of a single integer, the Euler characteristic $χ(K_n)$, equivalently the bouquet rank $b_n=χ(K_n)-1$.

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Fedor B. Lyudogovskiy. 2026-06-18. Local Complexity, Overlap Control, and Global Simplicity in Partition Graphs: A Conceptual Synthesis for G_n and K_n = Cl(G_n). https://arxiv.org/abs/2606.20773

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