Related papers: The gonality of chess graphs
In this paper we study queen's graphs, which encode the moves by a queen on an $n\times m$ chess board, through the lens of chip-firing games. We prove that their gonality is equal to $nm$ minus the independence number of the graph, and…
The gonality of a graph measures how difficult it is to move chips around the entirety of a graph according to certain chip-firing rules without introducing debt. In this paper we study the gonality of circulant graphs, a class of…
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 $P_{m \times n}(z)$…
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…
This paper aims to study the graph radii and diameters induced by the $k$-dimensional versions of the well-known six international chess pieces on every finite $\{n \times n \times \dots \times n\} \subseteq \mathbb{Z}^k$ lattice since they…
We prove new lower and upper bounds on the higher gonalities of finite graphs. These bounds are generalizations of known upper and lower bounds for first gonality to higher gonalities, including upper bounds on gonality involving…
In this paper we study Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve. We present an upper bound on the gonality of the Cartesian product of any two graphs, and…
Two dimensional rook graphs are the Cartesian product of two complete graphs. In this paper we prove that the gonality of these graphs is the expected value of $(n-1)m$ where $n$ is the size of the smaller complete graph and $m$ is the size…
We consider a variant of metrised graphs where the edge lengths take values in a commutative monoid, as a higher-rank generalisation of the notion of a tropical curve. Divisorial gonality, which Baker and Norine defined on combinatorial…
The bondage number b(G) of a graph G is the smallest number of edges of G whose removal from G results in a graph having the domination number larger than that of G. We show that, for a graph G having the maximum vertex degree $\Delta(G)$…
In how many ways can $n$ queens be placed on an $n \times n$ chessboard so that no two queens attack each other? This is the famous $n$-queens problem. Let $Q(n)$ denote the number of such configurations, and let $T(n)$ be the number of…
To any graph we associate a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic…
We provide lower bounds on the gonality of a graph in terms of its spectral and edge expansion. As a consequence, we see that the gonality of a random 3-regular graph is asymptotically almost surely greater than one seventh its genus.
We investigate multiple variants of the game Cops and Robbers. Playing it on an $n \times n$ toroidal chess graph, the game is varied by defining moves for cops and robbers differently, always mimicking moves of certain chess pieces. In…
The divisorial gonality of a graph is the minimum degree of a positive rank divisor on that graph. We introduce the multiplicity-free gonality of a graph, which restricts our consideration to divisors that place at most \(1\) chip on each…
In chomp on graphs, two players alternatingly pick an edge or a vertex from a graph. The player that cannot move any more loses. The questions one wants to answer for a given graph are: Which player has a winning strategy? Can a explicit…
In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the…
We prove that in the moduli space of genus-g metric graphs the locus of graphs with gonality at most d has the classical dimension min{3g-3,2g+2d-5}. This follows from a careful parameter count to establish the upper bound and a…
mim-width is a recent graph width measure that has seen applications in graph algorithms and problems related to propositional satisfiability. In this paper, we show linear lower bounds for the mim-width of strongly chordal split graphs,…
The $n$-Queens' graph, $\mathcal{Q}(n)$, is the graph associated to the $n \times n$ chessboard (a generalization of the classical $8 \times 8$ chessboard), with $n^2$ vertices, each one corresponding to a square of the chessboard. Two…