The gonality of chess graphs

Abstract

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 king's, bishop's, and knight's graphs, as well as for the toroidal versions of these graphs. We also prove that among all chess graphs, there exists an upper bound on gonality solely in terms of $\min\{m,n\}$, except for queen's, toroidal queen's, rook's, and toroidal bishop's graphs.

Figures (25)

• Figure 1: Legal king's moves from one vertex and corresponding edges on $4 \times 5$ king's graph, with dashed edges for the additional edges in the $4\times 5$ toroidal king's graph.
• Figure 2: The $3\times 4$ king's, bishop's, and knight's graphs; and the edges gained by the upper left vertex in the toroidal versions of those graphs.
• Figure 3: Two examples of set-firing moves, transforming a divisor $D$ into a divisor $D'$ and then into $D"$. The first set-firing move fires all vertices in the first column; the second set-firing move fires all vertices in the first two columns.
• Figure 4: An example of Dhar's Burning algorithm, starting at the vertex $q$.
• Figure 5: Two tree-cut decompositions of the $4\times 5$ grid. The first has width $16$; the second has width $4$.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• proof
• Remark 2.1
• Remark 2.2
• Lemma 2.1: §4 in debruyn2014treewidth, Theorem 1.1 in new_lower_bound
• Lemma 2.2: Proposition 3.1 in gonality_of_random_graphs
• Lemma 2.3: Corollary 3.2 in echavarria2021scramble
• Theorem 3.1
• proof