Queen Graphs: Movement, Domination, and Spectral Theory

• Queen Graphs are graph-theoretic models representing queen moves on chessboards and lattice boards, encapsulating row, column, and diagonal adjacencies.
• Researchers employ techniques like domination analysis, spectral theory, and edge-coloring to determine optimal configurations and computational thresholds.
• Extensions to toroidal, higher-dimensional, and polyomino regions illustrate the versatility of queen graphs in addressing complex geometric and combinatorial challenges.

Queen graphs are graph-theoretic models of queen movement on chessboards and related lattice boards. In the basic two-dimensional case, the vertex set is the set of board squares, and two vertices are adjacent exactly when the corresponding squares lie in a common row, column, or diagonal, so that a queen can move between them in one move. This family extends naturally to rectangular boards, toroidal boards, higher-dimensional boards, and constrained lattice regions such as polyominoes; across these settings, queen graphs have been studied through domination, independence, edge-coloring, chip-firing, spectral analysis, and exact computation (Bozóki et al., 2016, Ramani, 4 Apr 2026, Alpert et al., 2018, Morrison et al., 2023).

1. Definitions, variants, and geometric models

The ordinary queen graph on an $n\times n$ board is the graph whose vertices are the board squares and whose edges encode legal queen moves. In coordinate form, adjacency between $(i,j)$ and $(i',j')$ is given by one of

$$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$

This same definition appears in the rectangular queen’s graph $Q_{m\times n}$, where the board need not be square, and in the notation $Q_{m,n}$ used in edge-coloring work; in both cases the graph is the union of a rook graph and a bishop graph, with row/column edges supplied by rook moves and diagonal edges supplied by bishop moves (Bozóki et al., 2016, Jarnicki et al., 2016).

Several nonclassical variants are now standard. In the toroidal two-dimensional setting, the board is $\mathbb Z_n^2$, diagonal adjacency is interpreted modulo $n$, and the resulting graph is the toroidal queen graph. In higher dimension, the three-dimensional queen graph $Q_n^3$ has vertex set

$[n]^3=\{0,1,\dots,n-1\}^3,$

with adjacency determined by legal 3D queen moves along 13 undirected line families: 3 axis directions, 6 face diagonals, and 4 space diagonals. A toroidal 3D analogue replaces $(i,j)$0 by $(i,j)$1 and wraps those lines around the discrete torus (Ramani, 4 Apr 2026, Ramani, 4 Apr 2026).

A different generalization arises from restricted regions. For a polyomino $(i,j)$2, one may define a queen visibility graph $(i,j)$3 whose vertices are the tiles of $(i,j)$4, with adjacency whenever two tiles lie on a common row-, column-, or diagonal-line segment fully contained in $(i,j)$5. This is not the ordinary full-board queen graph, because holes, bottlenecks, and interrupted line of sight alter adjacency (Alpert et al., 2018). A further directed variant appears in Catalan queen-path models, where allowed moves are only east, north, and northeast by arbitrary positive distance, yielding a directed lattice graph rather than an undirected attack graph (Kung et al., 2011).

This diversity of definitions is not merely notational. Boundary effects, periodicity, and dimensionality change both local degree structure and global invariants. A plausible implication is that “queen graph” is best understood as a family of chess-move graphs rather than a single graph class.

2. Domination, coverage, and related extremal problems

Domination is one of the central themes in queen-graph theory. For a queen graph $(i,j)$6, a dominating set is a subset $(i,j)$7 such that every vertex lies in $(i,j)$8 or is adjacent to a vertex of $(i,j)$9. On rectangular boards, the domination number $(i',j')$0 has been computed for

$(i',j')$1

and the general lower bound

$(i',j')$2

was established. The same work showed that monotonicity can fail: for example,

$(i',j')$3

so enlarging the board can unlock more efficient geometric configurations rather than making domination harder (Bozóki et al., 2016).

The three-dimensional domination problem is substantially different. For the cubic board $(i',j')$4, the domination number satisfies

$(i',j')$5

from the maximum closed-neighborhood size, and also

$(i',j')$6

yielding

$(i',j')$7

The paper “On the Structure of 3D Queen Domination” identifies an inner core

$(i',j')$8

and studies

$(i',j')$9

the number of core vertices dominated by a queen at $$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$0. It proves a strict hierarchy by position type: $$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$1

$$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$2

with $$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$3 and $$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$4. In particular, no boundary queen matches the best inner-core coverage of an interior queen (Ramani, 4 Apr 2026).

Coverage-maximization with a fixed number of queens is distinct from domination minimization. In “Thresholds of Queen covers,” a $$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$5-configuration $$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$6 is evaluated by

$$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$7

and for sufficiently large $$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$8,

$$i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'.$$9

where

$Q_{m\times n}$0

Here $Q_{m\times n}$1 measures redundant attacks and $Q_{m\times n}$2 measures boundary truncation from off-center placement. The paper proves that for every fixed $Q_{m\times n}$3, there is a threshold beyond which every optimal $Q_{m\times n}$4-configuration is non-attacking, and a second threshold beyond which the set of optimal configurations stabilizes (Adhikari et al., 4 Aug 2025). This sharply separates maximum-coverage problems from minimum-domination problems: attacking queens may be useful on small boards, but asymptotically they become inefficient.

For polyominoes and polycubes, queen guarding is again a domination problem, now in the induced visibility graph. Every $Q_{m\times n}$5-tile $Q_{m\times n}$6-polycube can be guarded by at most $Q_{m\times n}$7 queens, and this bound is sometimes necessary; moreover, deciding whether a polyomino can be guarded by at most $Q_{m\times n}$8 queens is NP-hard (Alpert et al., 2018).

3. Independence, gonality, and nonattacking configurations

Independent sets in queen graphs are exactly nonattacking queen placements. On the ordinary square board, the classical $Q_{m\times n}$9-queens problem asks whether $Q_{m,n}$0, and the spectral survey of $Q_{m,n}$1 recalls the exact values

$Q_{m,n}$2

For rectangular boards $Q_{m,n}$3, the gonality paper uses the corresponding independence numbers

$Q_{m,n}$4

and for all other boards

$Q_{m,n}$5

Thus, except for the two smallest square exceptions, the maximum size of a nonattacking set is the smaller board dimension (Cardoso et al., 2020, Morrison et al., 2023).

A particularly strong connection between independence and chip-firing is the exact gonality formula

$Q_{m,n}$6

with the explicit specialization

$Q_{m,n}$7

The same identity holds for toroidal queen’s graphs: $Q_{m,n}$8 Moreover, maximum independent sets $Q_{m,n}$9 correspond bijectively to gonality-achieving divisor classes through

$\mathbb Z_n^2$0

and distinct maximum independent sets give distinct divisor classes (Morrison et al., 2023). In this setting, nonattacking queen placements are not merely combinatorial witnesses; they classify all minimal positive-rank divisors.

Toroidal independence becomes subtler in higher dimension. On $\mathbb Z_n^2$1, queens attack along every toroidal line whose direction vector lies in $\mathbb Z_n^2$2. The trivial upper bound is

$\mathbb Z_n^2$3

If $\mathbb Z_n^2$4 has no prime divisor less than $\mathbb Z_n^2$5, then this upper bound is attained: $\mathbb Z_n^2$6 But in dimension $\mathbb Z_n^2$7, if $\mathbb Z_n^2$8 is coprime to $\mathbb Z_n^2$9, divisible by $n$0, and not by $n$1, then

$n$2

For general fixed $n$3,

$n$4

so the trivial upper bound is asymptotically sharp up to order $n$5 (Williams, 2024).

A common misconception is that all large-board optimal configurations in fixed-$n$6 coverage problems should coincide with classical $n$7 nonattacking patterns. The fixed-$n$8 cover analysis shows otherwise. For $n$9, only three of the twelve classical fundamental solutions remain asymptotically cover-optimal when centralized on large boards, while for $Q_n^3$0 the single classical fundamental solution is never asymptotically cover-optimal (Adhikari et al., 4 Aug 2025). This suggests that independence is necessary in the asymptotic regime, but not sufficient to characterize optimal coverage.

4. Spectral and algebraic theory

The ordinary $Q_n^3$1-Queens’ graph $Q_n^3$2 is nonregular, with degree determined by peripheral layers. If $Q_n^3$3 denotes the $Q_n^3$4-th layer from the boundary, then every vertex $Q_n^3$5 has degree

$Q_n^3$6

Hence

$Q_n^3$7

and

$Q_n^3$8

The number of edges is

$Q_n^3$9

Despite this nonregularity, the least eigenvalue is uniformly bounded: $[n]^3=\{0,1,\dots,n-1\}^3,$0 and in fact $[n]^3=\{0,1,\dots,n-1\}^3,$1 is an eigenvalue of multiplicity $[n]^3=\{0,1,\dots,n-1\}^3,$2 for every $[n]^3=\{0,1,\dots,n-1\}^3,$3 (Cardoso et al., 2020).

The proof uses an edge clique partition by rows, columns, and both diagonal families. More generally, if $[n]^3=\{0,1,\dots,n-1\}^3,$4 is an edge clique partition of a graph $[n]^3=\{0,1,\dots,n-1\}^3,$5, and $[n]^3=\{0,1,\dots,n-1\}^3,$6 is the maximum number of cliques of $[n]^3=\{0,1,\dots,n-1\}^3,$7 containing a single vertex, then every eigenvalue $[n]^3=\{0,1,\dots,n-1\}^3,$8 of $[n]^3=\{0,1,\dots,n-1\}^3,$9 satisfies

$(i,j)$00

For $(i,j)$01, the natural partition gives $(i,j)$02, producing the bound $(i,j)$03. Equality is characterized by linear conditions requiring all row, column, and diagonal sums of an eigenvector to vanish, together with zero entries at the corners (Cardoso et al., 2020). This is one of the cleaner examples in which chessboard geometry directly controls a spectral extremum.

The same paper proves that $(i,j)$04 is also an eigenvalue of $(i,j)$05, with multiplicity at least

$(i,j)$06

It further gives an algorithm constructing an equitable partition of $(i,j)$07 with

$(i,j)$08

cells, so that quotient-matrix eigenvalues supply main eigenvalues of $(i,j)$09 (Cardoso et al., 2020).

A more recent decomposition writes the adjacency matrix of the $(i,j)$10-Queens’ graph as

$(i,j)$11

where $(i,j)$12 is the disjoint union of two triangular graphs $(i,j)$13 and $(i,j)$14, $(i,j)$15 is a disjoint union of cliques corresponding to one diagonal family, and $(i,j)$16 are disjoint unions of complete bipartite graphs corresponding to cross-triangle row and column interactions (Cardoso et al., 1 Aug 2025). The triangular graphs satisfy

$(i,j)$17

and are integral graphs. Their least eigenvalue is $(i,j)$18 with multiplicity

$(i,j)$19

for $(i,j)$20, while their full adjacency spectrum follows a consistent integer pattern with top eigenvalue $(i,j)$21 (Cardoso et al., 1 Aug 2025). Weyl’s inequalities are then used to derive lower and upper bounds on eigenvalues of $(i,j)$22.

The toroidal 3D case is spectrally more rigid because the graph is an abelian Cayley graph. For the toroidal three-dimensional queen graph $(i,j)$23 on $(i,j)$24, with $(i,j)$25 odd and $(i,j)$26, Fourier characters diagonalize the adjacency matrix and each frequency $(i,j)$27 has eigenvalue

$(i,j)$28

where $(i,j)$29 counts queen directions orthogonal to $(i,j)$30 modulo $(i,j)$31. The only possible values are

$(i,j)$32

with multiplicities

$(i,j)$33

$(i,j)$34

Equivalently, the adjacency eigenvalues are

$(i,j)$35

with the corresponding multiplicities above (Ramani, 4 Apr 2026). This toroidal spectrum is computed exactly, whereas the ordinary bounded 3D queen graph is dominated by boundary effects and does not admit such a translation-invariant analysis.

5. Edge-coloring and structural decomposition

The edge-chromatic number of the rectangular queen graph $(i,j)$36 is governed by the interplay between rook and bishop edges. Since

$(i,j)$37

with disjoint edge sets, one has

$(i,j)$38

The number of edges is

$(i,j)$39

Using Vizing’s dichotomy, the central question is whether $(i,j)$40 is Class 1 or Class 2 (Jarnicki et al., 2016).

The proved classification is extensive. The graph $(i,j)$41 is Class 1 if at least one of $(i,j)$42 is even, or if $(i,j)$43 and both are odd. It is also Class 1 for all odd $(i,j)$44. On the other hand, if $(i,j)$45 are odd and

$(i,j)$46

then $(i,j)$47 is overfull and therefore Class 2 (Jarnicki et al., 2016). The guiding conjecture is that this overfullness obstruction is the only one: $(i,j)$48 Extensive computation reported there supports this statement.

The methods are structurally specific to queen graphs. They combine optimal edge-colorings of rook graphs and bishop graphs, extremal bishop colorings, a “ladder coloring” of the rook graph, and a derived multicycle built from the canonical bishop coloring. The key transfer statement is that if the derived multicycle $(i,j)$49 can be edge-colored with $(i,j)$50 colors, then $(i,j)$51 is Class 1 (Jarnicki et al., 2016). This is a highly queen-specific reduction: the difficulty lies precisely in coordinating the diagonal part with the row-column part so that one bishop color can be absorbed into the rook palette.

A plausible implication is that queen graphs sit in an intermediate regime between product-like graphs and line-incidence graphs. They inherit enough decomposability from rook-plus-bishop structure to permit specialized arguments, but not enough regularity for those arguments to collapse to general theorems.

6. Exact computation, certified search, and open directions

Exact queen-graph computation has shifted toward proof-producing methods. In the classical 2D case, queen domination can be encoded as SAT with one Boolean variable per square and one domination clause per square,

$(i,j)$52

together with an at-most-$(i,j)$53 cardinality constraint

$(i,j)$54

Using modulo totalizer encodings, static $(i,j)$55-symmetry breaking through lex constraints, Cube-and-Conquer, and LRAT proof generation, “Queen Domination by SAT Solving” corrected a published discrepancy for $(i,j)$56 and resolved the previously open $(i,j)$57 case (Rostami et al., 16 Aug 2025). The number of non-isomorphic minimum dominating sets for $(i,j)$58 was completely certified, with

$(i,j)$59

non-isomorphic optimal solutions for $(i,j)$60 rather than the previously reported $(i,j)$61, and

$(i,j)$62

for $(i,j)$63 (Rostami et al., 16 Aug 2025). This episode is significant because it established verifiability as a methodological standard in queen-graph computation.

In the 3D domination setting, exact small-$(i,j)$64 values have been certified by integer linear programming: $(i,j)$65

$(i,j)$66

For $(i,j)$67, the reported bounds are

$(i,j)$68

The same work exploits the full octahedral symmetry group

$(i,j)$69

of order $(i,j)$70 and a fundamental domain

$(i,j)$71

showing that every dominating set has an $(i,j)$72-equivalent representative whose first queen lies in $(i,j)$73 (Ramani, 4 Apr 2026). This reduction is lossless and illustrates how structural theorems can feed directly into exact optimization.

Several open problems remain central. For ordinary rectangular domination, monotonicity is unresolved beyond the known failures, and the lower bound

$(i,j)$74

is not believed to be the final word (Bozóki et al., 2016). For 3D domination, no exact formula for $(i,j)$75 is known, the asymptotic constant in

$(i,j)$76

is unknown, and even $(i,j)$77 remains open (Ramani, 4 Apr 2026). For edge-coloring, the overfullness conjecture remains unresolved in the remaining odd-by-odd region (Jarnicki et al., 2016). For higher-dimensional toroidal independence, the exact criterion for attaining $(i,j)$78 nonattacking queens is unknown outside the currently proved cases (Williams, 2024). And in the fixed-$(i,j)$79 coverage problem, the sharp asymptotics of the non-attacking and stabilizing thresholds are not determined (Adhikari et al., 4 Aug 2025).

Taken together, these results portray queen graphs as a technically diverse family in which geometry, symmetry, arithmetic, and exact certification all matter. Their study links classical chessboard questions to domination theory, spectral graph theory, edge-coloring, chip-firing, SAT and ILP certification, Cayley-graph Fourier analysis, and incidence-geometric optimization.