Coloring discrete pseudomanifolds

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