Coloring discrete pseudomanifolds

TL;DR

This paper establishes new bounds on the chromatic number of discrete d-pseudomanifolds, improving understanding of their coloring properties under various structural conditions.

Contribution

It introduces three main results providing bounds on the chromatic number for different classes of d-pseudomanifolds, including improved and optimal bounds.

Findings

Chromatic bounds for any d-pseudomanifold: d+1 to 2d+2

Improved bound for pseudomanifolds as Zykov joins: ≤ 2d+1

Optimal bound for joins with even-cycle factors: ≤ ⌈3(d+1)/2⌉

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 $d$.