Coloring discrete pseudomanifolds

TL;DR

The paper generalizes the chromatic theory of geometric graphs from manifolds to discrete $d$-pseudomanifolds, proving the universal bound $d+1 \le X(K) \le 2d+2$ for any connected $d$-pseudomanifold and deriving sharper bounds in structured cases such as $K=S^k+K'$. It leverages graph duality and operations like the Zykov join and Cartesian product to study colorings, showing that $X(G+H)=X(G)+X(H)$ and that even-cycle spheres often admit smaller chromatic numbers, $X(S^k)=2\lceil (k+1)/2\rceil$, while odd-cycle spheres can be larger, $X(S^k)=3\lceil (k+1)/2\rceil$. When the pseudomanifold decomposes with an even-cycle spherical factor and the dimension $k$ is near $d$, the authors establish the near-optimal bound $X(K) \leq \lceil 3(d+1)/2\rceil$, highlighting how structural constraints tighten color requirements. Collectively, these results extend discrete geometric coloring theory, provide a coherent framework for bounding chromatic numbers of pseudomanifolds, and illuminate how spherical factors influence tightness of bounds in higher dimensions.

Abstract

This paper presents three main results on coloring discrete $d$-pseudomanifolds: $(1)$ the general chromatic bounds $d+1 \leq X(K) \leq 2d+2$ for any $d$-pseudomanifold $K$; $(2)$ an improved bound $X(K) \leq 2d+1$ for pseudomanifolds expressible as a Zykov join $K = S^k + K'$; $(3)$ the optimal bound $X(K)\leq\lceil 3(d+1)/2\rceil$ under the additional assumptions that the spherical join factor $S^k$ is built from even-cycles and its dimension $k$ is close to $d$.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof