Papers

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra over $\mathbb{C}$ and let $L(\rho)$ be the irreducible integrable $\mathfrak{g}$-module with highest weight $\rho$. Let $I$ be a subgraph of the Dynkin diagram of $\mathfrak{g}$ which has only simple bonds and no cycle of length $\geq 3$. For every subset $D$ of $I$, denote by $\beta_D$ the sum of the simple roots corresponding to $D$. To every $D \subset I$ such that $\lambda_{D,I} = 2\rho - \beta_I - \beta_D$ is dominant, we associate certain elements $\pi_{D,I}$ of weight $\lambda_{D,I} {-} \rho$ in the crystal $B(\rho)$, which depend on the choice of a root vertex in each connected component of $I$. Then we prove that our elements are $\rho$-dominant elements of $B(\rho)$, hence provide new families of components of the tensor product $L(\rho)\otimes L(\rho)$.

Erasure codes are a core component of most existing large-scale distributed storage systems, ensuring reliability against node failures. Recent work has shown that adapting code parameters to changing node failure rates can lead to significant storage savings. The default approach is to re-encode the data under a new code, which consumes substantial system resources. Code conversion was introduced to reduce this cost. However, existing work has mainly focused on conversions within specific classes of codes. In this paper, we study scalar linear code conversion in the merge regime for arbitrary linear codes. We derive universal lower bounds on the write and read costs in terms of unchanged and read symbols. The bounds are refined using generalized Hamming weights, which capture support-growth properties of subcodes and can give sharper estimates than minimum-distance-only arguments. We show that the framework recovers known bounds for important special cases and can be strictly stronger when the final code has nontrivial jumps in its generalized Hamming weight hierarchy. We then apply the framework to Reed-Muller codes and construct explicit Reed-Muller convertible codes using the Plotkin decomposition. For a natural Reed-Muller parameter regime, the construction attains the derived write-cost lower bound. For the read cost, the generalized-Hamming-weight analysis is sharp for one initial block, while a gap remains for the other block.

Signed total Roman domination is a variant of the classic Roman domination-problem in graphs. A signed total Roman dominating function (STRD function) on a graph $G=(V,E)$ is a function $f: V \to \{-1,1,2\}$ such that (i) $\sum_{u \in N(v)} f(u) \geq 1$ for all $v \in V$, where $N(v)$ denotes the neighborhood of $v$, and (ii) every vertex $v$ with $f(v) = -1$ is adjacent to a vertex $u$ with $f(u) = 2$. The weight of $f$ is $\sum_{v \in V} f(v)$. The signed total Roman domination number of $G$ is the minimum weight among all its STRD functions. A signed total Roman dominating family (STRD family) on $G$ is a family $\{f_1, \ldots, f_d\}$ of pairwise distinct STRD functions such that $\sum_{i=1}^{d} f_i(v) \leq 1$ for all $v \in V$. The signed total Roman domatic number of $G$ is the maximum size among all its STRD families. In this paper, we relate the signed total Roman domination number of a cubic graph to its open packing number, 2-tuple total domination number, and signed total domination number, allowing us to derive sharp bounds on the first invariant and to establish new $\mathcal{NP}$-completeness results for all four invariants. We demonstrate that having a degree-3 vertex determines a graph's signed total Roman domatic number. Combined with known results this implies that the associated decision problem is easy for graphs with maximum degree at most three and $\mathcal{NP}$-complete otherwise. To contrast these general hardness results, we determine signed total Roman domination and domatic numbers in complete multipartite graphs. Despite their simple structure, this is a non-trivial task.

For a graph $G$, a proper $k$-coloring of $G$ is equitable if the sizes of any two color classes differ by at most one. The \textsc{Equitable $k$-Coloring} problem asks, for a given graph $G$ and integer $k$, whether $G$ admits an equitable $k$-coloring. Bodlaender and Fomin showed that it is polynomial-time solvable on graphs of bounded treewidth, while it remains $\NP$-hard on cographs, and thus on graphs of constant clique-width. Fellows et al. showed that the problem becomes $\mathsf{W[1]}$-hard when parameterized by tree-width (and hence clique-width) plus the number of colors~$k$. We first show that, for every fixed $k$, counting equitable $k$-colorings is polynomial-time solvable on graph classes of bounded clique-width, given a clique-width expression. We then show that, under $\mathsf{SETH}$, the dependence on clique-width in this algorithm is essentially optimal. As a consequence, our results provide a fairly tight picture of the complexity of \textsc{Equitable $k$-Coloring} with respect to the combined parameter $k$+clique-width. Second, we refine our clique-width algorithm for the linear setting. We show that there exists an algorithm, given an integer $k\ge 1$ and an $n$-vertex graph $G$ together with a linear $w$-expression constructing $G$, computes the number of equitable $k$-colorings of $G$ in time $\max\{1,2^k-2\}^w\cdot n^{k+O(1)}$. Third, we consider a different structural restriction, namely the class of $P_t$-free graphs. A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. This is a different setting from bounded clique-width; in particular, already $P_5$-free graphs have unbounded clique-width. Nevertheless, we show that for every $P_t$-free graph $G$, the number of equitable list $3$-colorings of $G$ can be computed in subexponential time.

In their 2010 article entitled “How to invert $n$ cups $m$ at a time" in Mathematics Today, Man-Keung Siu and Ian Stewart extend a classic trick in which $3$ cups are flipped $2$ at a time to any number of cups, flipped any number at a time. Here, we generalize another feature of the problem. A typical cup in our universe, as far as we know, has two “sides", right side up and upside down. But what if the cup had $k$ “sides" with $k\ge 2$? We call these $k-$hypercups. The classic trick depends on parity. In this article, we show the analogous hypercups trick depends on greatest common divisors. We generalize all of Siu and Stewart's results to $k-$hypercups. Our results imply the known results for $2-$hypercups, also known as cups.

A graph $G$ with maximum degree $\Delta$ is $\Delta$-critical if it is connected, satisfies $\chi'(G)=\Delta+1$, and the deletion of any edge reduces its chromatic index to $\Delta$. A $\Delta$-critical graph $G$ is called nontrivial if it contains no $\Delta$-overfull subgraph; that is, no $H \subseteq G$ such that $|E(H)| > \Delta \lfloor |V(H)| /2 \rfloor$. There are no $1$-critical graphs, and the $2$-critical graphs are exactly the odd cycles. By work of Chetwynd and Yap from 1983, there is a unique nontrivial $3$-critical graph of order $9$, and exactly two nontrivial $3$-critical graphs of order $11$. In 2005, Bokal, Brinkmann, and Grünewald showed that there are exactly fourteen nontrivial $3$-critical graphs of order $13$. For even orders, Brinkmann and Steffen proved in 1997 that no $3$-critical graphs of even order exist below order $22$, there is exactly one 3-critical graph of order $22$, and that there are exactly nine $3$-critical graphs of order $24$. To the best of our knowledge, there has been no further progress on the existence of nontrivial $3$-critical graphs of odd orders beyond the cases established two decades ago. In this paper, using computer-assisted search techniques, we determine the exact numbers of nontrivial $3$-critical graphs of odd orders from $15$ to $21$. The same data pipeline also reproduces the known order-$22$ count of Brinkmann and Steffen, with one nontrivial $3$-critical graph. Beyond enumeration, we prove a characterization theorem for all nontrivial 3-critical graphs, with one case based on snarks. We also provide an analysis of our search algorithm.

The Maker-Breaker sabotage game is played on a graph $G$ by Runner and Blocker. They play in turns, Runner first moves along a not yet traversed edge from her current position, afterwards Blocker removes one edge. The goal of Runner is to visit as many vertices of $G$ as possible, Blocker's goal is opposite. Assuming that both players use optimal strategies, the number of vertices visited by Runner determines an invariant called the sabotage number ${\rm sab}(G)$ of $G$. A formula for the sabotage number of an arbitrary tree is proved which can be evaluated in polynomial time. For a unicyclic graph $G$ it is proved that ${\rm sab}(G)\in \{{\rm sab}^-(G), {\rm sab}^-(G)+1\}$, where ${\rm sab}^-(G)$ is the lower sabotage number of $G$. The sabotage number of a bridgeless subcubic graph is sharply bounded from the above by the maximum girth. The sabotage number is also bounded for complete bipartite graphs and generalized Sierpiński graphs, and determined exactly in some special cases.

In the pinwheel scheduling problem, each task $i$ is associated with a positive integer $a_i$ called its period, and we want to (perpetually) schedule one task per day so that each task $i$ is performed at least once every $a_i$ days. An obvious necessary condition for schedulability is that the density, defined as the sum of execution rates $1/a_i$, does not exceed $1$. We prove that all instances with density not exceeding $5/6$ are schedulable, as was conjectured by Chan and Chin in 1993. Like some of the known partial progress towards the conjecture, our proof involves computer search for schedules for a large but finite set of instances. A key idea in our reduction to these finite cases is to generalize the problem to fractional (non-integer) periods in an appropriate way. As byproducts of our ideas, we obtain a simple proof that every instance with two distinct periods and density at most $1$ is schedulable, as well as a fast algorithm for the bamboo garden trimming problem with approximation ratio $4/3$.

Extending a little-known construction of Ruzsa, we characterize the quadruples $(s,t,u,v)$ of nonnegative integers such that the inequality $|sA-tA| \leq |uA-vA|$ holds for all finite sets $A \subseteq \mathbb{Z}$

We study the discrete space-time wave equation on a $(q+1)$-regular graph $X$ associated with the affine Laplace-type operator. For the forward time-difference scheme we derive explicit formulas for the two fundamental solutions (wave kernels) in terms of discrete modified Bessel functions and the non-backtracking walk counts on $X$ thus providing a direct and explicit link between wave propagation and combinatorial graph data. Utilizing uniqueness property of the wave kernel, we prove a new trace-type formula associated to the affine Laplace-type operator on $X$ and apply it to deduce many combinatorial identities. For example, we derive a closed-form expression for evaluation of some trigonometric sums twisted by an additive character as well as evaluations of finite sums of Chebyshev polynomials twisted by binomial coefficients.

The concept of graph capacity extends graph span by considering the maximum number of agents that can simultaneously traverse a graph while preserving a prescribed minimum distance. We study the Cartesian $1$-capacity, corresponding to the movement in which exactly one agent moves at each step. We establish general lower and upper bounds based on structural graph properties and derive exact results for trees. Our work highlights the role of branching and bridge structures in determining the Cartesian $1$-capacity and provides new insights into collision-free multi-agent motion on graphs.

The game of Cops and Robbers is a two player pursuit game on graphs where a team of cops attempts to catch a robber. The cop number $c(G)$ of a graph $G$ is the minimum number of cops needed to guarantee a winning strategy in $G$. A famous conjecture of Meyniel says that if $G$ is connected, then $c(G)=O(\sqrt{n})$. Erde, Kang, Lehner, Mohar and Schmid considered its generalization to $k$-uniform hypergraphs and conjectured that the cop number of such hypergraphs is $O(\sqrt{n/k})$. This may be understood as a hypergraph version of Meyniel's conjecture. In this paper we prove this conjecture for a class of expanding hypergraphs and show that with high probability the conjecture holds for random hypergraphs $H^k(n,p))$ provided $k\geq \log^3n$ and the typical degree, $p\binom{n-1}{k-1}$, is $\omega(\log^3n)$.

The aim of this paper is to establish critical duality theory for ratios of nonnegative homogeneous convex functions (shorten for RC functions) and differences of convex functions (abbreviated as DC functions) on Banach spaces. Specifically, we establish a series of duality results on critical point theory and Morse theory for RC functions, including the homotopy type of sublevel sets, the Morse critical points and their Rothe critical groups, Lagrange critical points and their multiplicities, Lusternik-Schnirelman min-max critical values, Poincare polynomials, as well as the structure of handlebody decompositions, all of which are proved to be preserved under polarity dual. Moreover, we obtain the first critical duality theory of DC functions which does not depend on the DC decomposition. This answers a question left open from the work of Toland on DC functions and the work of Le-Pham on DC programming. We apply these results to provide a reformulation of the graph Cheeger constant using zonotopes; we introduce the contact data which serves as a geometric characterization of Lagrange criticality; and we show that the eigenproblems for 1-Laplacian and $\infty$-Laplacian on hypergraphs are equivalent to the contact problems of zonotopes, which indeed establishes a new characterization of zonotopes. We also prove a duality equivalence for certain nonlinear eigenvalue problems and bifurcation problems. Our study here reveals an intricate interaction of critical point theory with other fields such as convex analysis, combinatorial geometry, and nonlinear eigenproblems on graphs.

We prove the nonnegativity and palindromicity of a family of polynomials arising from $q$-Hoggatt numbers. The nonnegativity is derived from Stanley's $(P,\omega)$-partition theory through a standard Young tableau formula, while the palindromicity is proved by an involution on rectangular standard Young tableaux. Our result confirms a conjecture of Cigler.

We introduce avalanche homology as a new (di)graph homology theory, based on the dynamics of the sandpile model. Avalanche homology is the simplicial homology of the avalanche complex generated from the sets of unstable vertices at the time steps of the sandpile dynamics. In this work we focus on digraphs, and our main results give the homotopy types of the avalanche complex for directed paths and directed cycles for certain initial configurations of the sandpile dynamics. Even for such simple digraphs a wide range of topologies can arise, and we compare this to the directed flag complex and to the recently introduced burning homology. Furthermore, the dynamics yields very naturally a filtered simplicial complex, and hence persistent avalanche homology.

Motivated by the twin homomorphism problem for Coxeter groups and the corresponding Hecke monoids, we find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.

Recent results established the maximum forest number $f(B)$ for balanced bipartite graphs under Ore-type degree sum conditions. In this paper, we extend these results by determining the exact value of the maximum forest number as a closed-form formula for general bipartite graphs under Ore-type conditions, answering an open question posed by Yu. We prove that the maximum forest number is bounded by a discrete optimization over at most six critical structural coordinate points dictated by hyperbolic density constraints. Furthermore, we establish that deciding whether a specific balanced bipartite graph on $2n$ vertices has a forest number of at least $n+2$ is NP-complete. This implies that while the maximum possible forest number can be exactly bounded, computing the exact forest number for a given graph remains computationally intractable, and a simple structural characterization via finite forbidden induced subgraphs cannot exist unless P = NP.

The supersaturation problem asks, for a fixed $r$-graph $\mathcal F$, for the minimum number of copies of $\mathcal F$ in an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges. Mubayi conjectured a local form of supersaturation under a stability hypothesis: if $\mathcal F$ is non-$r$-partite and stable, meaning roughly that the extremal $\mathcal F$-free construction is unique and all near-extremal $\mathcal F$-free $r$-graphs are close to it, then this minimum should be at least $q c(n,\mathcal F)$, where $c(n,\mathcal F)$ is the minimum number of copies created by adding one edge to the extremal $\mathcal F$-free $r$-graph. We disprove this conjectured local lower bound in every uniformity. For every $r\ge2$ and every $K>1$, we construct a stable $r$-graph $\mathcal F$ such that, for all sufficiently large $n$ and every $1\le q\le \delta n$, there is an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges and at most $K^{-1}q c(n,\mathcal F)$ copies of $\mathcal F$. Thus the conjectured lower bound can already fail at $q=1$, and the failure can be by an arbitrarily large constant factor in every uniformity.

Let $G$ and $H$ be finite simple graphs and assume that either both are undirected or both are directed. We introduce and study the ideal of weak graph homomorphisms $I_{G\to H}$. We characterize all graphs $G$ and $H$ for which every (equivalently, some) power of $I_{G\to H}$ has a linear resolution. Moreover, unmixedness, Cohen-Macaulayness, projective dimension and Castelnuovo-Mumford regularity of these ideals are studied.

In network vulnerability analysis, it is crucial to evaluate the robustness of $k$-cores against vertex removals. A $k$-core is often fragile since removing a few vertices can trigger a large reduction in the core size, a phenomenon known as core collapse. In this paper, we study the problem of enumerating all minimal removable sets (MinRSs) of a given $k$-core, where a MinRS is a minimal nonempty set of vertices whose removal results in a smaller $k$-core graph. We consider this problem within a general mathematical framework based on monotone systems. We show that, for a monotone system that is given with an underlying graph $G=(V,E)$, all MinRSs of a solution can be enumerated in $O((n+m)n\tau_\omega)$ time, where $n=|V|$, $m=|E|$ and $\tau_\omega$ denotes the computation time of evaluating the monotone function of the system. Furthermore, if the system satisfies the newly defined in-dominating seed property, the complexity drops to $O((n+m) \log n \cdot \tau_\omega)$ time. We prove that standard $k$-cores in undirected graphs satisfy this property, enabling MinRS enumeration in $O((n+m)\log n)$ time, a significant improvement over the baseline. We also extend our framework to enumerate all solutions in a given monotone system. This yields an $O((n+m)\log n)$-delay algorithm for all $k$-core subgraphs, outperforming an algorithm given by [Boley et al., Theoretical Computer Science, 2010]. Our framework is applicable to various $k$-core extensions, including weighted $k$-cores, multi-layer $\boldsymbol{k}$-cores, and $(k,\ell)$-cores.

A matching covered graph is a brick if it is 3-connected and bicritical. An edge of a matching covered graph G is a forcing edge if it lies in precisely one perfect matching of G. We prove that every vertex of a simple brick G is incident with a forcing edge if and only if G is an odd wheel or is isomorphic to one of the following three graphs: the triangular prism \overline{C_6},\overline{C_6}^+ or the bicorn H_8. Here \overline{C_6}^+ is obtained from \overline{C_6} by adding an edge between two nonadjacent vertices.

There are many formulas in the literature providing roots of a real cubic that avoid some of the well-known pathologies of Cardano's formulas. Among these, we identify two that consistently provide the unique roots of a depressed cubic that have the greatest and smallest absolute value, whenever those exist. We call these the longest and shortest roots. The existence conditions are elementary and are in terms of the signs of the coefficients and the discriminant. Our proofs use two algebraic identities satisfied by hypergeometric functions; once the standard real branches are fixed, the root comparisons are entirely real. As an application, the longest-root formula gives an explicit factorization of all but a vanishing proportion of depressed real quartics.

We extend graph theoretic characterizations of time structurable extensive games from finite informational digraphs to countable and continuum digraphs built on continuum rooted real trees. Introducing the symmetric class quotient poset of information sets as the canonical object an external clock must order, we develop a theory of transitive reduction for these quotients and prove existence and uniqueness under an interval finiteness condition. Under natural topological regularity assumptions like closed reachability classes, order density, and sigma compactness, we give necessary and sufficient conditions for real valued time labelings and construct measurable selection procedures in Polish spaces. We also provide constructive algorithms and a differential game example illustrating applications and computational implications.

It is well known that every regular Cayley map $M = \CM(G,X,p)$ on a finite group $G$ with respect to an inverse-closed generating set $X$ of $G$ and a specified cyclic permutation $p$ on $X$ corresponds to a skew morphism $\varphi$ on $G$ such that the restriction of $\varphi$ to $X$ is $p$. The skew-type of the map $M$ is defined as the index $[G:\Ker \varphi]$, which equals the number of distinct values in $\mathbb{Z}_{|\varphi|}$ taken by the associated power function $\pi$ of the skew morphism $\varphi$. In this paper, we develop a covering theory of skew morphisms and as an application we provide a classification of regular Cayley maps of skew-type three on the semidihedral groups.

The classical existence result of Nash guarantees that every finite noncooperative game admits an equilibrium in mixed strategies, but it leaves open the question of when pure strategy equilibria exist. This paper develops a structural approach to that question by exploiting properties of the best-response correspondence on finite strategy sets. Building on recent work, we derive new sufficient conditions for the existence of pure strategy Nash equilibria in finite games. We introduce several broad classes of finite games for which pure equilibria are guaranteed, including a class of generalized potential games that generalizes unilaterally competitive games and a class characterized by the existence of an aggregate-payoff maximizer over an ordered set. Our results clarify the role of acyclicity, and aggregation in producing pure equilibria and connect disparate sufficient-condition results in the literature into a unified framework.

In 2017, Suda, Tanaka and Tokushige conjectured that if $1>p_1\ge\cdots\ge p_n>0$ with $p_3\le \frac{1}{2}$, then every intersecting family $\mathcal A\subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le p_1$, where $\mu_{\mathbf{p}}$ is the non-uniform product measure defined by $\mu_{\mathbf{p}}(\mathcal{A})=\sum_{A\in\mathcal{A}} \prod_{i\in A} p_i \prod_{j\in [n]\setminus A}(1-p_j)$. In addition, if $p_1 > p_3$ or $p_1 < \frac{1}{2}$, then equality holds if and only if $\mathcal{A}$ is a star centered at some $i \in [n]$ with $p_i = p_1$. In this paper, we prove this conjecture in the following stronger $t$-intersecting form: for any $t\ge 1$, if $p_{t+2}\le \frac{1}{t+1}$, then every $t$-intersecting family $\mathcal{A} \subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le \prod_{i=1}^t p_i$. Moreover, when $p_{t+2}<\frac{1}{t+1}$, equality holds if and only if $\mathcal{A}=\{A\subseteq [n]: T\subseteq A\}$ for some $T\in \binom{[n]}{t}$ with $\prod_{i\in T} p_i=\prod_{i=1}^t p_i$. Our result unifies and generalizes the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut.

The Grone–Merris conjecture, proved by Bai in~2011, states that the spectrum of the graph Laplacian $\Delta_0 = -\operatorname{div}\operatorname{grad}$ is majorized by the conjugate of the vertex degree sequence. Duval and Reiner proposed a simplicial complex analogue of this statement. On a graph, where triangles serve as $2$-simplices, their conjecture reduces to the assertion that the spectrum of $\operatorname{curl}^*\operatorname{curl}$ is majorized by the conjugate of the second-order degree sequence, which records the number of triangles containing each vertex. We prove that the sum of the two largest eigenvalues of $\operatorname{curl}^*\operatorname{curl}$ does not exceed the sum of the first two entries of that conjugate sequence. This confirms the first two majorization inequalities predicted by Duval and Reiner for $\operatorname{curl}^*\operatorname{curl}$. As a corollary, we obtain upper bounds for the two largest eigenvalues of the full graph Helmholtzian $\Delta_1 = -\operatorname{grad}\operatorname{div} + \operatorname{curl}^*\operatorname{curl}$. The same result extends to the up-Laplacian of any $3$-family, yielding a concrete step towards the Duval–Reiner conjecture in dimension~$1$.

Topological indices are key quantitative descriptors in mathematical chemistry, unchanged under symmetry operations and retaining graph connectivity; they capture molecular structural features to provide insights into molecular stability and chemical properties, becoming indispensable in cheminformatics and theoretical chemistry. Among degree-based indices, the Sombor index is widely concerned for capturing structural information, and motivated by enhanced structural discrimination, the augmented Sombor index ($ASO$) is defined for a connected graph $\Omega$ with $|V(\Omega)|\geq 3$ as $$ASO(\Omega) = \sum_{v_iv_j\in E(\Omega)} \sqrt{\frac{{d_i^2 + d_j^2}}{d_i + d_j - 2}},$$ where $d_i$ and $d_j$ are the degrees of vertices $v_i$ and $v_j$, respectively. Within the scope of this study, we first establish several sharp bounds for the augmented Sombor index and characterize the extremal graphs attaining these bounds. In particular, we determine the minimum value of the $ASO$ index for unicyclic graphs with a prescribed girth and characterize all graphs achieving this minimum. We also identify the second maximum $ASO$ value among trees and characterize the corresponding extremal tree structures. Furthermore, the minimum and maximum values of the $ASO$ index for bipartite graphs and chemical graphs are obtained, together with a complete characterization of the associated extremal graphs. In addition, we characterize the chemical trees that maximize the $ASO$ index. The chemical applicability of the $ASO$ index is investigated through quantitative structure-property relationship (QSPR) analysis, supported by a comparative assessment of several variants of the Sombor index. Finally, we present concluding remarks and outline potential directions for future research on the augmented Sombor index of graphs.

We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from $O(n^3)$ to $O(n^2)$. To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly $1.8 n$ for grids of size $82 \le n \le 119$. For the Smallest Complete Set problem, we find configurations of size roughly $0.95 n$, providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

Completing partial Latin squares is NP-complete. Ryser characterized precisely when an $r\times s$ Latin rectangle can be completed to an $n\times n$ Latin square. We extend this theorem to a broader setting. Let $M$ be an $n\times n$ array whose top-left $r\times s$ subarray is filled with symbols from $\{1,2,\dots,k\}$, where $1\leq k\leq n^2$. We determine necessary and sufficient conditions under which the remaining cells of $M$ can be filled so that each symbol $\ell\in\{1,2,\dots,k\}$ appears exactly $\rho_\ell$ times in total, while the numbers of occurrences of each symbol in any two rows and in any two columns differ by at most one. Equivalently, our result characterizes when a partial edge-coloring of $K_{r,s}$ can be extended to an edge-coloring of $K_{n,n}$ in which each color class is spanning and almost regular, with prescribed color-class sizes. In this sense, our theorem generalizes Baranyai's construction of almost-regular colorings of complete uniform multipartite hypergraphs, restricted to the bipartite case. Our theorem unifies and generalizes several classical results. When $s=k=\rho_1=\cdots=\rho_k=n$, it reduces to Hall's theorem, and when $k=\rho_1=\cdots=\rho_k=n$, it yields Ryser's completion theorem for Latin rectangles. Additional special cases recover results of Goldwasser, Hilton, Hoffman, and Özkan, as well as a theorem of Bahmanian. Thus, our work may be viewed as a common generalization of these completion theorems and Baranyai's theorem.