Papers

We have two dices which have $n$ labels. We want to assign a pair of labelings with positive integers to them such that the sums of the labels on them are the same and have the same distribution as at the standard labeling. We call such a pair of labelings a (dice) game of size $2$ (with dice size $m$). Here we give a lower and an upper bound for the dice games of size $2$ with Sicherman dice of dice size $p^{k}$ where $p$ is a prime. We use the standard technique of generating polynomials to prove a non-closed combinatorial formula.

The zero forcing number of a graph $G$, $Z(G)$, is a well-studied parameter which arises from a color changing process and has strong connections to minimum rank, critical ideals and related invariants. In this work, we consider the complementary parameter $\mz(G) = |V(G)| - Z(G)$. This parameter is monotone under taking induced subgraphs. This leads us to the study of graphs for which $\mz(G)$ is bounded, via forbidden induced subgraphs. We prove that the number of minimal forbidden graphs for graphs with $\mz(G)\leq k$ is finite for any $k\geq 1$. We determine the complete set of minimal forbidden graphs for the case $k = 3$, and we provide partial characterizations of graphs with $\mz(G) \leq 3$, based on girth. Our results suggest new directions for the structural understanding of zero forcing-type parameters.

Given two tournaments of order $n$, Bollobás and Scott defined their discrepancy as the largest deviation of their overlap from its random average under relabelling, and they asked whether the resulting discrepancy is always $\Omega (n^{3/2})$. We answer this question by proving that there is an absolute constant $c>0$ such that every pair of tournaments $T,U$ of order $n$ has discrepancy at least $cn^{3/2}$. More generally, if $D$ and $H$ are oriented graphs of order \( n \) with \( e(D) = p \binom{n}{2} \) and \( e(H) = q \binom{n}{2} \) satisfying $16/n \leq p, q \leq 1 - 16/n$, then there is an absolute constant \( c > 0 \) such that their discrepancy is at least $c(p(1-p)q(1-q))^{3}n^{3/2}$. We also show that these two-graph estimates extend to intersections of any fixed number of graphs, tournaments, and oriented graphs.

We give a uniform realization of Kirillov-Reshetikhin crystals $B^{r,s}$, in terms of PBW crystals, for all affine types, $s \geq 1$, and nodes $r$ in the orbit of the $0$ node in the Dynkin diagram. Our proof is almost uniform except we prove some technical lemmas by first proving the case when the $r$-th fundamental weight $\Lambda_r$ is minuscule by using properties of minuscule crystal and then extend this to the case when $\Lambda_r$ is cominuscule using virtual crystals.

The Khintchine inequality provides a lower bound on the expected absolute value of a weighted sum of independent Rademacher random variables. In the classical setting, when the weight vector has unit norm, this lower bound is a constant, with equality attained only for a simple family of extremal configurations. A refined version due to De, Diakonikolas, and Servedio (2013) – referred to as the linear Khintchine inequality – strengthens this by establishing a lower bound that depends linearly on the distance of the weight vector from the extremal set. In this paper, we present a refined analysis of this dependence on the weight vector. Our results reveal a phase transition in the rate of improvement: when the dimension exceeds six, the lower bound undergoes an abrupt change as the weight vector deviates from the minimizer. Additionally, we improve the slope constant in linear Khintchine inequality. As a consequence, we establish an improved lower bound on the degree-1 Fourier weight for linear threshold functions $\mathbf{W}^{\leq 1}[\mathrm{LTF}] \geq 0.53317$, marking progress towards a conjecture of O'Donnell.

Eigenspaces of hypercube graph adjacency matrices are well understood. We extend this to subgraphs obtained by removing a vertex or antipodal pair of vertices, motivated by a question on zonotopes.

We study deep loci for the reduced-word Berenstein-Fomin-Zelevinsky cluster atlas on type-A double Bruhat cells. As defined in CGSS24, for a seed collection on a cluster variety X, the deep locus is the complement of the union of all corresponding cluster tori. We focus on double Bruhat cells G(u,v) for G = SL(n). Our main results are several theorems describing the structure of these deep loci, including their interaction with the Poisson structure, the relationships between the deep loci of different cells, and a tracking formula that allows us to compute the reduced-word deep locus of a cell from a lower-dimensional one. The paper also includes explicit low-rank calculations for double Bruhat cells in SL(3) and Borel double Bruhat cells in SL(4). Finally, we discuss the relationship between these results and some general properties of deep loci.

We determine the Grundy total domination number and the skew zero forcing number for every Cartesian product of two paths or cycles, thereby closing the nonmatching bounds previously known for these rectangular, cylindrical, and toroidal families. For $2\le a\le b$, \[ Z_-(P_a\square P_b) =a-\mathbf1_{\{a\ {\rm odd},\,b\ {\rm even}\}}. \] For $p\ge2$ and $c\ge3$, \[ Z_-(P_p\square C_c)= \begin{cases} \min\{p,c\},&c\text{ is odd}, \min\{2p,c\},&c\text{ is even}. \end{cases} \] Finally, for $3\le a\le b$, \[ Z_-(C_a\square C_b)= \begin{cases} 2a-1,&a=b\text{ odd}, 2a,&a=b\text{ even}, \min\{b,2a\},&a<b,\ a\text{ odd}, a,&a<b,\ a\text{ even},\ b\text{ odd}, 2a,&a<b,\ a,b\text{ even}. \end{cases} \] In every case, $\gamma_{\mathrm{gr}}^t(G)=|V(G)|-Z_-(G)$. The path-containing lower bounds use Kronecker differences of skew-symmetric or hollow symmetric factor matrices, and they also determine maximum skew nullity and minimum skew rank. They further produce an infinite family for which maximum skew nullity is strictly smaller than skew zero forcing. The new cycle–cycle lower bounds use a cyclic column-defect estimate, applied to complete columns when the shorter cycle is odd and, after a one-sided bipartite reduction, to half-columns when both cycles are even. Explicit forcing constructions give matching upper bounds throughout.

We determine the complete eigenvalue region of the following Markov chains on $2q$ states, for $q\ge2$. The states are $A_0,\ldots,A_{q-1}$ and $B_0,\ldots,B_{q-1}$. Each transition $A_j\to A_{j+1}$ and $B_j\to B_{j+1}$, for $0\le j<q-1$, has probability one. From $A_{q-1}$, the chain moves to $A_0$ with probability $a$ and to $B_0$ with probability $1-a$. From $B_{q-1}$, it moves to $B_0$ with probability $b$ and to $A_j$ with probability $(1-b)p_j$. Here $a,b\in[0,1]$ and $p_j\ge0$ with $\sum_{j=0}^{q-1}p_j=1$; these parameters vary over all permitted values. Every segment from zero to an attainable eigenvalue lies in the region. Its unit-circle points are exactly the roots of unity of order at most $2q$, as for unrestricted stochastic matrices of the same order, yet the full region is strictly smaller. It excludes $0<|\operatorname{Arg}\lambda|<\pi/q$. For odd $q$, the final nonreal boundary arc ends at a negative real point inside the unit disk; the remaining interval to $-1$ is also boundary. Every transition matrix has the same characteristic polynomial as one with $a=b$. Among $q+1$ fixed matrices in this balanced family, convex combinations of at most three realize every attainable eigenvalue, and two suffice on the boundary. Both bounds are sharp within this set. Consecutive reduced fractions in $[1/(2q),1/2]$ with denominators at most $2q$ select the upper nonreal boundary constructions. Each boundary radius is the unique solution of an equation whose left-hand side increases strictly with the radius.

Let $T_r(n)$ denote the complete $r$-partite graph on $n$ vertices whose part sizes differ by at most one. A graph $H$ is called $K_{r+1}$-Turán-good if, for all sufficiently large $n$, the graph $T_r(n)$ contains the maximum number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs. We say that $H$ is $K_{r+1}$-Turán-stable if, for every $\varepsilon>0$, there exist $\delta>0$ and $n_0$ such that, whenever $n\ge n_0$ and an $n$-vertex $K_{r+1}$-free graph $G$ contains at least $\mathrm{ex}(n,H,K_{r+1})-\delta n^{v(H)}$ copies of $H$, the edit distance between $G$ and $T_r(n)$ is at most $\varepsilon n^2$, where $v(H)$ denotes the order of $H$. In this paper, we prove that every graph $H$ is both $K_{r+1}$-Turán-good and $K_{r+1}$-Turán-stable for $r\ge 4v(H)^3+11v(H)^2$. This strengthens results of Morrison, Nir, Norin, Rzążewski, and Wesolek~[J. Combin. Theory Ser. B, 2023] and Gerbner and Hama Karim~[J. Graph Theory, 2024], and gives a positive answer to a question of Morrison, Nir, Norin, Rzążewski, and Wesolek. We also prove that $\inj(H,G)\le \inj(H,T_r(n))$ for every $n$-vertex $K_{r+1}$-free graph $G$ and for $r\ge 40v(H)^3$, where $\inj(H,G)$ denotes the number of injective homomorphisms from $H$ to $G$. Finally, we give a negative answer to another question of Morrison, Nir, Norin, Rzążewski, and Wesolek.

Connectivity can turn separate contributions into a public good that benefits participants and nonparticipants alike. Yet only participants pay for it, and each can avoid her charge by withdrawing while enjoying whatever public output survives. How do network connections constrain who can or must pay, and how large a cost can the group bear? We study a participation game in which public output equals the size of the largest connected group of active players, whose members divide a fixed operating cost. The output a participant destroys by withdrawing is her withdrawal responsibility and bounds her payment. These bounds determine the group's cost-bearing capacity and, whenever the group can be kept active, which participants must pay. We prove that among connected networks with at least five players, the path uniquely maximizes the cost-bearing capacity of a group, yet minimizes the expected surviving output after a uniformly selected participant withdraws. As the group grows, the ratio of maximal capacity to gross social benefit under full participation converges to one quarter. Fragility can therefore strengthen voluntary finance, but even the most financeable network cannot close the gap between the value it creates and what its participants will pay to sustain it.

A resolving set in a graph is a set of landmarks whose distance vectors distinguish all vertices. We use information theory to prove lower bounds for metric dimension and class dimension in distance-regular graphs and association schemes arising from finite geometry. The core idea is that, for a fixed landmark, a random object usually lies in one overwhelmingly likely distance or relation class. For classical dual polar graphs, with rank and type fixed and $q\to\infty$ through the admissible field orders, we prove $\mu(\Gamma(q,d,e))=\Theta_{d,e}(q^e)$ for $d\geq 2$ and $e>0$. The lower bound uses opposition as the typical distance. For the upper bound, we take, for each of a constant number of $(d-1)$-dimensional singular subspaces, all generators containing it. For Grassmann graphs, bilinear forms graphs, and attenuated-space schemes, we obtain lower bounds of the same exponential order as the known incidence constructions.

We study path-width and the closely related parameter line-width in countably infinite graphs. Our first result characterises the graphs of finite path-width: they are the graphs that do not have infinitely many vertices of infinite degree, do not have infinitely many pairwise disjoint infinite paths, and contain no subdivision of some finite tree of maximum degree 3. We then investigate universality under the subgraph relation for graphs of bounded path-width or line-width. In particular, we prove that there exists a universal graph with line-width $\mathcal{O}(k^2)$ for the class of graphs with line-width at most $k$. In contrast, we show that no graph of finite path-width is universal for the class of locally finite graphs with path-width $1$. Finally, we show that for each $k\geq 2$, every universal graph for the class of graphs with path-width at most $k$ has line-width at least $k + 1$.

The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove that $D(n)\ll n^{3/4}$. Second, we prove for almost all positive integers $n$ that $D(n)\ll_B n^{1/2}(\log n)^B$ for every $B>5/6$, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that $D(n)\ll_{\varepsilon} n^{\varepsilon}$. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.

A majority C-coloring of a graph $G$ assigns colors to the vertices such that every vertex shares its color with at least half of its neighbors. The maximum number of colors that can be used in such a coloring of $G$ is denoted by $\overline{\chi}_{\geqslant}(G)$. In this paper, the focus is on the majority C-coloring in Cartesian product graphs. It is shown that $\overline{\chi}_{\geqslant}(G \square H) \ge \overline{\chi}_{\geqslant}(G) \overline{\chi}_{\geqslant}(H)$ gives a sharp lower bound, but the difference also can be arbitrarily large. For two-dimensional Hamming graphs, the exact value $\overline{\chi}_{\geqslant}(K_m \square K_n) = \min\{m,n\}$ is established. Balanced Hamming graphs of higher dimension, that is the $k$th powers of complete graphs with respect to the Cartesian product, are also studied. It is proved that $\overline{\chi}_{\geqslant}(K_n^{\square, k})= n^{k/2}$ holds for every even integer $k$. If $k$ is odd and the Hamming graph is the $k$-dimensional hypercube, then $\overline{\chi}_{\geqslant}(K_2^{\square, k})= 2^{\lfloor k/2\rfloor}$. On the other hand, a majority C-coloring of $K_n^{\square, k}$ with at least $3 n^{\lfloor k/2\rfloor}/2 $ colors is presented for every $n \ge 7$ and odd $k \ge 3$. For Cartesian grids, the main result shows that $\overline{\chi}_{\geqslant}(P_m \square P_n) = 1 + \lfloor m/2\rfloor \lfloor n/2\rfloor$ if at least one of $m$ and $n$ is odd, while $\overline{\chi}_{\geqslant}(P_m \square P_n)=mn/4$ holds if both parameters are even and $m \ge n \ge 4$. The paper concludes with a conjecture and several open problems.

A graphical design is a subset of vertices of a graph, along with a weight for each chosen vertex, that can perfectly average chosen subspaces of functions on the graph. A design is uniformly weighted if all the weights are equal, and several well-known combinatorial objects such as orthogonal arrays, combinatorial block designs and t-wise permutations are uniformly weighted graphical designs. While one might expect to see uniformly weighted designs in structured graphs, they do not always exist. In this paper we characterize the existence of uniformly weighted graphical designs, and use our result to provide several families of graphs that have, and do not have, such designs. Our results offer a polyhedral view of the structures that control the existence and cardinalities of these designs. In particular, we characterize all uniformly weighted designs of threshold graphs, and provide a geometric proof of the duality of linear codes and linear orthogonal arrays. We also provide a novel construction for graphs whose Laplacian characteristic polynomials are almost irreducible, to produce families without uniformly weighted designs.

For a tree T, let gamma(T) be its domination number and let mu(T) count the Laplacian eigenvalues in [0,1). The strict inequality gamma(T)/mu(T)<4/3 leaves open three linked questions: the exact finite-order defect, the structure near equality, and whether a fixed defect permits only finitely many local obstructions. We answer all three. For every n>=2, max_{|V(T)|=n} (7 gamma(T)-9 mu(T)) = floor((n-20)/9). Thus order twenty-nine is the first at which the ratio 9/7 can be exceeded. The proof is based on an exact decomposition of Phi(T)=|V(T)|-20-9(7 gamma(T)-9 mu(T)) into nonnegative integer terms, together with a clean-contraction identity. For subcubic trees, equality cases are canonical lifts of trees with perfect matchings. We classify every layer through Phi=42, locating successive phase transitions at 20, 21, 40, 41, and 42. The finiteness mechanism is simple to state: after the repeatable matched-skeleton pieces are removed, fixed slack leaves only a bounded exceptional core. Formally, a local compactness theorem bounds every nonordinary weighted deep component at fixed packing surplus and residual negative index. Consequently, for each fixed k, every subcubic tree with Phi(T)=k consists of a bounded ported defect kernel, an arbitrary compatible forest of ordinary tiles, and at most floor(k/21) clean expansions. All unbounded statements are proved symbolically; exact computation is used only for explicitly bounded atom lists and independent finite verification.

We give a polynomial-time algorithm to compute the number $b_n$ of separable derangements of $[n]$. This algorithm is based on a generating function technique which tracks permutations along with their occupied diagonals, where each permutation is counted once for every such diagonal. We provide bounds for the proportion of separable permutations which are derangements, show that $b_n$ and the large Schröder numbers have the same exponential growth constant $(3 + 2 \sqrt{2})$, and use the first 3000 terms to conjecture more explicit asymptotic behavior. This partially answers several questions about separable derangements recently posed by Vatter.

Let $s\geq 2$ and let $v_1,\ldots,v_{2^{s-1}}\in F_2^s\setminus\{0\}$ have sum zero. The prescribed-difference matching problem asks whether $F_2^s$ can be partitioned into pairs whose differences, counted with multiplicity, are exactly these vectors. Fix a hyperplane $H\leq F_2^s$. An $h$-hole instance is one in which exactly $h$ prescribed differences lie in $H$, counted with multiplicity. We prove that every four-hole instance has a solution for $s\geq 3$ and every eight-hole instance has a solution for $s\geq 4$. Thus no restriction on the multiplicities or on the sum of the holes is needed beyond the global zero-sum condition. After the internal pairs are placed, their endpoints are deleted from the two affine halves determined by $H$, and the remaining vectors must be paired with the prescribed crossing differences. A locally valid placement of the internal pairs need not admit such a completion. We encode the possible completions by coefficients of signed determinants and use the Walsh transform to sum over all disjoint legal placements while keeping the crossing profile fixed. Nonvanishing of this sum guarantees that at least one placement extends to a full matching. The four-hole and zero-sum eight-hole theorems are proved theoretically. The general eight-hole proof uses three finite exact verifications, independent of the crossing profile: a local filter calculation, a classification of the remaining eight-hole configurations, and integer certificates for those configurations. The verification software, exact inputs, and recorded outputs are archived in a versioned Zenodo record.

We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector \(x\) of active features is encoded through an overcomplete dictionary \(W\), and feature recovery is performed by applying \(\operatorname{ReLU}(W^\top W x+b)\) with an appropriate bias vector \(b\). We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order \(d/\log n\). In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with \(n>d+1\), we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization—which should be of independent interest to frame theorists—of the distribution of signs in the Gram matrix.

We consider the following problem: let $n>k$ — positive integers, $G$ — a graph on $n$ vertices (undirected, without loops or multiple edges). Let $h_k(G)$ denote the number of unordered pairs of vertices of the graph $G$ whose degrees differ by less than $k$. We seek to determine the smallest possible value $f(n,k)$ of $h_k(G)$. The interest in this question is motivated by the fact that the bipartite analogue of the problem allowed S. Cichomski and F. Petrov \cite{CP} to prove the Burdzy–Pitman conjecture on the spread of independent identically distributed random variables.

We prove that the $q$-Ehrhart series of a hyperplane slice of a cube is a rational function with an explicit denominator that satisfies $q$-reciprocity, confirming a conjecture of Reiner and Rhoades for these polytopes. To do this, we find a generating set for the orbit harmonics ideal, which also yields the Hilbert series and graded Frobenius characteristic of the associated quotient. We further show that the harmonic algebra of a hypersimplex $\Delta$ is generated as an algebra by the harmonic space of $\Delta$ using structural results on 2-factors of regular multigraphs. In particular, the harmonic algebra is finitely generated, giving a second proof of rationality.

Let $q(n)$ denote the largest number of queens that can be placed on an $n\times n$ chessboard so that no queen attacks more than one other queen. We prove that $q(n)=\lfloor4n/3\rfloor$ for every $n\geqslant6$, and that $q(n)=n$ for $n\leqslant5$, which settles a previously conjectural value. As a corollary, we also settle that, in the version of the problem where each queen attacks exactly one other queen, the answer is $2\lfloor2n/3\rfloor$, again as previously conjectured.

A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an $\mathfrak{sl}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, so the boundary data of such a stranding is recorded by a row-strict tableau. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis. This non recursive construction recovers Fontaine's bases and produces others. For $\mathfrak{sl}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we ask whether any two strandings of a web are connected by a sequence of local moves called strand reversals. We answer this question affirmatively for $\mathfrak{sl}_3$, extending a known result for closed webs to webs with boundary; in particular, a leading term stranding of any $\mathfrak{sl}_3$ web is always reachable from an arbitrary stranding.

The Non-Cancelling Intersections (NCI) conjecture of Amarilli, Monet and Suciu [arXiv:2401.16210] states that the union of a finite family of sets can always be built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In Wilhelm [arXiv:2608.19414] the conjecture was shown to fail when the witnessing dot-algebra expression is required to be left-linear. Here we remove that restriction and show that the conjecture is false in general: there is a finite lattice admitting no dot-algebra representation of its top element whatsoever. The counterexample is a lattice $P_{p,\mathfrak{m}}$ as in Wilhelm [arXiv:2608.19414], and the argument differs in only two ways. First, we replace the sequential "toggle game" of Wilhelm [arXiv:2608.19414] by a corresponding tree-shaped object, the plane tree, which stands to dot-algebra trees as the toggle game stands to left-linear ones. Second, we use a marked plane in which there is no admissible set of any size between $2p$ and $4p$, which also removes the need for the Erdős–Beck theorem and for the arithmetic Nullstellensatz. Consequently $p$ need not be astronomically large: every prime $p \ge 10^{5}$ works.

The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. In each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy guaranteeing a red copy of $G$ or a blue copy of $H$ within $t$ rounds. For every fixed integer $k\ge4$, the best-known lower bounds for $\tilde{r}(K_{1,k},P_n)$ and $\tilde{r}(K_{1,k},C_n)$ are $\left(\frac{k+3}{4}+o(1)\right)n$ as $n\to\infty$. We improve the corresponding asymptotic upper bounds from $(k+o(1))n$ to $\left(\frac{2k+4}{5}+o(1)\right)n$ as $n\to\infty$.

Erdős and Gallai in 1959 proved the seminal result that every $n$-vertex graph with no cycle of length at least $2t+2$ has at most $\tfrac{2t+1}{2}(n-1)$ edges. We prove the extension that, for every sufficiently large $t$, the same quantity is also the sharp extremal bound for graphs with no $t$ consecutive even cycle lengths, resolving a conjecture of Verstraëte. Thus, at the Erdős-Gallai threshold, forcing an entire interval of even cycle lengths costs no more than forcing its longest member. More precisely, every $n$-vertex graph $G$ with $e(G)\ge \tfrac{(2t+1)(n-1)}2$ $\bullet$ either contains $t$ consecutive even cycle lengths, $\bullet$ or equality holds and $G$ is connected with every block isomorphic to $K_{2t+1}$. As consequences, for every sufficiently large even $k$ we determine the sharp edge thresholds forcing a cycle of length $0\pmod k$ or $2\pmod k$, answering questions of Bai, Grzesik, Li, and Prorok and of Gao, Li, Ma and Xie, respectively, for sufficiently large even $k$. The proof develops a stability-enhanced sublinear expander method. Its main new ingredient is a dense-case decomposition that recovers the lengths lost in the expander extraction by combining a flexible dense core with rooted cycle families in the vertices outside the core.

For every natural number $r\geq 2$, we construct $r^{th}$ powers of bipartite graphs whose chromatic number is quadratic in their clique number, showing that the straightforward quadratic upper bound is best possible. We thereby settle an open problem posed by Chakraborty, Chandran, Jacob and Pillai [J. Graph Theory 112(3) (2026), 235-254] by establishing the sharpness of the quadratic bound for squares of bipartite graphs.

The zero divisor graph $\Gamma(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $\Gamma(R)$ for two classes of finite commutative rings: reduced rings and quotients of principal ideal domains. Our proofs use a new combinatorial gadget, the integral covering graph, that captures the structure shared by both ring families and generalizes the disjointness graph on subsets of $[n]$, where two subsets are adjacent if and only if they are disjoint. In doing so, we answer two questions recently posed by L.~Sunil Chandran and Suraj Kumar Sahoo in Boxicity of Zero Divisor Graphs, Discrete Applied Mathematics 391 (2026).

We extend the conjecture of Erdős and Gyárfás on monochromatic path covers to the setting of monochromatic cycle covers. We prove that, for all $n$, every 2-edge-coloring of the complete graph on $n$ vertices contains a collection of at most $\lceil\sqrt{n}\rceil$ monochromatic cycles, all of the same color, that together cover all vertices. The order of the bound is best possible, and the ceiling is necessary for infinitely many $n$.