Minimal simplicial degree $d$ self-maps of $\mathbb{S}^{n-1}\times \mathbb{S}^1$

The degree of a map between orientable manifolds is a fundamental concept in topology, providing important information about the structure of manifolds and the behavior of maps between them. A simplicial cell complex $K$ is called a \emph{colored triangulation} of a closed PL $n$-manifold $M$ if the $1$-skeleton of $K$ admits a proper vertex-coloring with $n+1$ colors and $|K|$ is PL-homeomorphic to $M$.
In this article, we construct, for every $d \in \mathbb{Z}$ and $n \geq 2$, a degree $d$ simplicial map from a $(2(n+1)\max\{|d|,1\})$-facet colored triangulation of $\mathbb{S}^{n-1} \times \mathbb{S}^1$ to the standard $2(n+1)$-facet colored triangulation of $\mathbb{S}^{n-1} \times \mathbb{S}^1$. Additionally, for every $d \in \mathbb{Z}$ and $n \geq 2$, we construct a degree $d$ simplicial map from a $(2\max\{|d|,1\})$-facet colored triangulation of $\mathbb{S}^n$ to the standard $2$-facet colored triangulation of $\mathbb{S}^n$.
For $M = \mathbb{S}^{n-1} \times \mathbb{S}^1$ and $\mathbb{S}^n$, with $n \geq 2$, these simplicial degree $d$ self-maps of $M$ are minimal with respect to their standard colored triangulations, in the sense that there does not exist a colored triangulation $K$ of $M$ with fewer facets than the constructed one that admits a simplicial map $f : K \to K'$ of degree $d$, where $K'$ denotes the standard colored triangulation of $M$.