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Chess graphs encode the moves that a particular chess piece can make on an $m\times n$ chessboard. We study through these graphs through the lens of chip-firing games and graph gonality. We provide upper and lower bounds for the gonality of…
A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to attack each other. (Recall that a queen can attack anything on the same row, column, or diagonal as itself.) This problem is…
The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks,…
The function that counts the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is a piecewise polynomial which is eventually a…
The present paper aims to extend the knight's tour problem for $k$-dimensional grids of the form $\{0,1\}^k$ to other fairy chess leapers. Accordingly, we constructively show the existence of closed tours in $2 \times 2 \times \cdots \times…
A placement of chess pieces on a chessboard is called dominating, if each free square of the chessboard is under attack by at least one piece. In this contribution we compute the number of dominating arrangements of $k$ rooks on an $n\times…
Automatic chess problem or puzzle composition typically involves generating and testing various different positions, sometimes using particular piece sets. Once a position has been generated, it is then usually tested for positional…
We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that…
The classic Rock-Paper-Scissors game of size 3 and its extension, Rock-Paper-Scissors-Lizard-Spock, are modeled by directed graphs called tournaments. They can be further extended to any odd size. The extended games are regular tournaments…
The $m \times n$ king graph consists of all locations on an $m \times n$ chessboard, where edges are legal moves of a chess king. %where each vertex represents a square on a chessboard and each edge is a legal move. Let $P_{m \times n}(z)$…
The unit-distance graph on the $n$-dimensional integer lattice $\mathbb{Z}^n$ is called the $n$-dimensional grid. We attempt to maximize the girth of a $k$-regular (possibly induced) subgraph of the $n$-dimensional grid, and provide…
The study of combinatorial games is intimately tied to the study of graphs, as any game can be realized as a directed graph in which players take turns traversing the edges until reaching a sink. However, there have heretofore been few…
We study outer length billiards; our main results are as follows. We prove 3- and 4-periodic versions of the Ivrii conjecture. We show that, for every period $n\ge 3$, there exists a functional space of billiard tables that possess…
The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set $M$ containing pairs of cells such that each cell of the $m_1…
How many mutually non-attacking queens can be placed on a d-dimensional chessboard of size n? The n-queens problem in higher dimensions is a generalization of the well-known n-queens problem. We provide a comprehensive overview of…
In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$…
Cops and robbers is a pursuit-evasion game played on graphs. We completely classify the cop numbers for $n \times n$ knight graphs and queen graphs. This completes the classification of the cop numbers for all $n \times n$ classical chess…
The metric dimension of a graph is the minimum number of landmark vertices required so that every vertex can be uniquely identified by its distances to the landmarks. This parameter captures the fundamental tradeoff between compact…
The Queen's Domination problem, studied for over 160 years, poses the following question: What is the least number of queens that can be arranged on a $m \times n$ chessboard so that they either attack or occupy every cell? We propose a…
This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…