On the maximum number of non attacking rooks on a high-dimensional simplicial chessboard

The simplicial rook graph ${\rm \mathcal{SR}}(m,n)$ is the graph whose vertices are vectors in $\mathbb{N}^m$ such that for each vector the summation of its coordinates is $n$ and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. Martin and Wagner (Graphs Combin. (2015) 31:1589--1611) asked about the independence number of ${\rm \mathcal{SR}}(m,n)$ that is the maximum number of non attacking rooks which can be placed on a $(m-1)$-dimensional simplicial chessboard of side length $n+1$. In this work, we solve this problem and show that $\alpha({\rm \mathcal{SR}}(m,n))=\big(1-o(1)\big)\frac{\binom{n+m-1}{n}}{m}$. We also prove that for the domination number of rook graphs we have $\gamma({\rm \mathcal{SR}}(m, n))= \Theta (n^{m-2})$. Moreover we show that these graphs are Hamiltonian. The cyclic simplicial rook graph ${\rm \mathcal{CSR}}(m,n)$ is the graph whose vertices are vectors in $\mathbb{Z}^{m}_{n}$ such that for each vector the summation of its coordinates modulo $n$ is $0$ and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. In this work we determine several properties of these graphs such as independence number, chromatic number and automorphism group. Among other results, we also prove that computing the distance between two vertices of a given ${\rm \mathcal{CSR}}(m,n)$ is $\mathbf{NP}$-hard in terms of $n$ and $m$.