Papers

In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of $\mathbb Z_{pq}$, where $p$ and $q$ are distinct odd primes with $q>p$. We first recall the natural partition of the zero-divisor graph into the two prime classes and then give a detailed characterization of those subsets of $BS(\Gamma(\mathbb Z_{pq}))$ that form metric bases when $q\geq 2p-1$. The proof is expanded by separating the role of closed neighborhoods, rows of subdivision vertices, and forbidden twin configurations. We then investigate $M$-subdivision graphs obtained by subdividing selected edges of $\Gamma(\mathbb Z_{pq})$. In addition to the lower bounds for subdivisions of $p-3$ and $p-2$ edges, we prove an exact formula for matching subdivisions of arbitrary size $r$, $0\leq r\leq p-2$, namely $\dim(G_r)=p+q-r-4$. Several consequences are included to illustrate how a small matching subdivision can reduce the localization cost of the original zero-divisor network.

There are four covering numbers $\mathrm{c}_g^{\mathcal{G}}(H),\mathrm{c}_u^{\mathcal{G}}(H),\mathrm{c}_l^{\mathcal{G}}(H),\mathrm{c}_f^{\mathcal{G}}(H)$, each of which measures in a slightly different way how well the edges of a graph $H$ (called a host) can be covered with graphs of a class $\mathcal{G}$ (called a guest class). If we require the graphs of $\mathcal{G}$ to correspond to induced subgraphs of $H$, we obtain an induced variant $\mathrm{ic}_x^{\mathcal{G}}$ for each covering number $\mathrm{c}_x^{\mathcal{G}}$ which satisfies $\mathrm{c}_x^{\mathcal{G}}(H) \leq \mathrm{ic}_x^{\mathcal{G}}(H)$ for every graph $H$. Yet, in general $\mathrm{ic}_x^{\mathcal{G}}$ cannot be bounded in terms of $\mathrm{c}_x^{\mathcal{G}}$. If there exists for a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ a function $f$ such that $\mathrm{ic}_x^{\mathcal{G}}(H) \leq f(\mathrm{c}_x^{\mathcal{G}}(H))$ for every graph $H \in \mathcal{H}$, we call $f$ a binding function. Within this work, we study for which structural properties of a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ such binding functions exist. We consider guest classes $\mathcal{G}$ that are monotone, hereditary, component-closed or neither, and have bounded maximum average degree, bounded chromatic number or neither. The host classes $\mathcal{H}$ we consider have bounded treewidth, exclude some minor, have bounded maximum average degree, bounded chromatic number, or none of these properties. For $219$ out of the $240$ possible $3$-tuples of properties for $\mathcal{G}$ and $\mathcal{H}$ and covering numbers we either provide a binding function or an example where no such function exists. In particular, we show that such binding functions always exist for hereditary guest classes $\mathcal{G}$ of bounded maximum average degree for three of the four covering numbers, but may not for the fourth kind.

Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In this paper, we consider the vertices that are in every minimum identifying code in a graph. There are two types of such vertices: always-forced vertices that belong to all identifying codes (minimum or not) and min-forced vertices that belong to all minimum identifying codes. A vertex is called proper-min-forced if it is min-forced but not always-forced. We show an upper bound $2n/3$ for the number of such proper-min-forced vertices in a closed-twin-free graph of order $n$. Moreover, for integers $n$ divisible by three, we construct an infinite family of graphs in which there are $2n/3-1$ such vertices. In addition, we determine the maximum number of edges in a graph of even order such that the graph contains proper-min-forced vertices. We also show that the decision problem of determining whether a given vertex in a graph is proper-min-forced is co-NP-hard.

Ramanujan's modular equations of degrees $3$, $5$, $7$, $11$ and $23$ are closely related to certain theta function identities. Warnaar generalized the identities arising from the modular equations of degrees $3$ and $7$ to a general theta function identity. Kim subsequently obtained a further general theta function identity associated with the modular equations of degrees $5$, $11$ and $23$, and established several general theta function identities containing known partition theorems as special cases. In this paper, we present an elementary approach to Kim's general theta function identities, yielding $q$-series proofs of these identities within a common framework.

For a graph $H$, the $H$-freeness edge-deletion problem is the algorithmic problem of finding, for an input graph $G$, the minimum number of edges of $G$ whose deletion turns $G$ into an $H$-free graph. We show that for every graph $H$ containing a cycle, this problem is NP-hard. This proves a conjecture of Gishboliner, Levanzov and Shapira, and completes the characterization of the complexity of the $H$-freeness edge-deletion problem, answering a question of Alon, Shapira and Sudakov.

We construct quantum Latin squares with maximal cardinality and quantum Latin hypercubes with maximal cardinality over the real field.

Makeev [2] stated that every finite Borel measure in $\mathbb R^d$ assigning zero mass to hyperplanes can be cut by $d$ mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary proof. The main ingredient is a Borsuk–Ulam theorem for $O(k)$: every $B_k$-equivariant map from $O(k)$ to a natural representation of the hyperoctahedral group $B_k$ has a zero. An explicit model map has one free orbit of zeros, consisting of the signed eigenbases of a diagonal operator with simple spectrum. A derivative computation and mod-$2$ equivariant degree complete the proof. This is a self-contained proof of a special case of the general Stiefel-manifold theorem in [5].

We prove the three symmetric Kanade-Russell identities modulo nine.

We provide a combinatorial description of the stabilization index of the $v$-function of powers of edge ideals for arbitrary graphs.

An $n\times n\times n$ array on $n^2$ symbols is a layer-rainbow Latin cube if every layer contains every symbol exactly once. We call it symmetric if $L_{ij\ell}=L_{j\ell i}=L_{\ell ij}$ for distinct $i,j,\ell$ and $L_{iij}=L_{jji}$, $L_{iji}=L_{jij}$, $L_{ijj}=L_{jii}$ for distinct $i,j$. We determine exactly when a symmetric layer-rainbow Latin cube of order $m$ embeds in one of order $n$, giving a three-dimensional analogue of Cruse's embedding theorem. Call a positive integer admissible if it is congruent to $0$ or $2$ modulo $3$, with $1$ admissible and $3$ excluded. For $n>m$, an embedding exists if and only if $m,n$ are admissible, $(m,n)\ne(2,5)$, and \[ \begin{cases} n\geq2m,&n-m\not\equiv1\pmod3,\\[1mm] \displaystyle n\geq m+\frac{\sqrt{48m^2+1}-1}{6},&n-m\equiv1\pmod3. \end{cases} \] Via the equivalent one-factorization problem for a non-uniform hypergraph, fair detachment reduces the proof to an exact integer allocation. We also determine the structure forced at both sharp boundaries and obtain infinitely many equality cases. At order eight, we construct a symmetric layer-rainbow Latin cube admitting the natural diagonal action of $\operatorname{PSL}(2,7)$, whose induced action on the $64$ symbols has orbit sizes $1,7,28,28$.

We show that if $\varphi$ is a colour-permuting automorphism of a connected, finite Cayley graph, and the order of the Cayley graph is either odd or square-free, then $\varphi$ is the composition of a group automorphism and a colour-preserving graph automorphism. Some analogous results are also established for isomorphisms between two different Cayley graphs.

We prove that if $\varphi$ is an isomorphism between two connected Cayley graphs of abelian groups, and $\varphi$ respects the natural edge-colourings of the Cayley graphs, then $\varphi$ is the composition of a group isomorphism and a colour-preserving graph automorphism. This implies that if every colour-preserving automorphism of a connected abelian Cayley graph $Cay(G;S)$ is an affine map, then the same is true for every colour-permuting automorphism. We also show that this property holds if and only if the subgroup generated by $\{\, s \in S \mid 2s \neq c \,\} \cup \{c\}$ has index $\le 2$ for every element $c$ of order $2$ in $G$.

For a finite real alphabet $T$, let $g(T)$ be the supremum of the lengths of words over $T$ containing no two adjacent nonempty blocks of equal length and equal sum. We classify all four-element alphabets with $g(T)<139$. Up to affine equivalence, these are the known endpoint-sum family and seven isolated classes. Exhaustive finite searches give the isolated values, while 77 explicit words prove that every remaining real alphabet admits a word of length 139. The latter argument uses exact intersections of finite unions of lines, so it covers irrational parameters as well. The cutoff is sharp because $g(\{0,2,3,7\})=139$. We also obtain a sharp cutoff of 167 within the family $\{0,1,x,x+2\}$, $x>1$. The required supplement prints the word certificates and specifies the complete-search evidence.

For every $d\geq2$, we prove that $d-1$ independent finite tiles in $\Z^d$ admit a fully periodic joint co-tile whenever they admit a joint co-tile. Here independence means that the tiles contain the origin and every choice of one nonzero vector from each tile is linearly independent. This removes property $(\star)$ from the periodic existence theorem of Meyerovitch, Sanadhya and Solomon and recovers Bhattacharya's theorem when $d=2$. More generally, we prove periodic existence for any finite family admitting a joint co-tile whose indicator is a fully periodic real function plus finitely many bounded real functions, each invariant under a subgroup of rank $d-1$. Extending the Boolean normal-form method of Greenfeld and Tao, we obtain a configuration in the translation orbit closure whose restrictions to cosets of a finite-index lattice are either invariant under a rank-$(d-1)$ subgroup or given by a three-term affine fractional-part formula with density one half. The tiling equations pair the restrictions given by this formula and determine a finite bipartite graph. A two-coloring replaces those restrictions by constants while preserving every tiling equation. We deduce a characterization of periodic tilability by independent companions and a decision algorithm for families containing an independent $(d-1)$-subfamily, with the dimension as part of the input.

For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we provide an alternative technique to establish that $q(G) \leq 4$ for any threshold graph $G$ as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further

OpenAI recently announced a proof of the Cycle Double Cover (CDC) Conjecture. Most media reports have characterized it as a 50-year-old open problem. In reality, according to a 1987 letter from Tutte to Fleischner, the Cycle Double Cover Problem has been open for at least 80 years. Two early results regarding the CDC conjecture were established in one of Tutte's 1949 publications.

For all positive integers $n,k$, we prove that the layer-refined antichain polynomial of the product poset $[2]\times[n]\times[k]$ is real stable. Jacobi-polynomial interlacing further shows that its diagonal specialization, the ordinary antichain polynomial of the same poset, has only simple, strictly negative zeros. For the special family $[2]\times[m]\times[m+1]$, explicit reciprocal identities give palindromicity; reciprocal pairing of the simple negative zeros then shows that every coefficient in the gamma expansion is strictly positive. Thus we prove Conjecture 4.3 of Ding and Dong and resolve all parts of their Conjecture 4.5, while strengthening its stated gamma-positivity consequence. The enumerative input is an explicit first-crossing reflection for two lattice paths, specialized from work of Krattenthaler and Sulanke.

We give an explicit finite sum for the number of ideals of the Lie algebra of strictly lower triangular $n\times n$ matrices over $\mathbb F_q$, valid for every prime power $q$. The sum runs over integer compositions, with weights expressed using ordinary and Gaussian binomial coefficients. A contraction bijection transforms Gagnon's configuration sum into a weighted enumeration of nonnesting partitions, with antichains of intervals chosen independently in each block. We derive a Stieltjes continued fraction for the block weights. Lagrange inversion and the coefficient formula for Stieltjes–Rogers polynomials then yield the explicit sum.

Monge matrices arise throughout combinatorial optimization and algorithm design; the Monge polytope $\mathcal{M}_{pq}$ is the set of $p \times q$ Monge matrices lying inside the standard simplex on the set of matrix coordinates. We find a Stanley decomposition of the associated affine semigroup, and use it to obtain a volume formula for $\mathcal{M}_{pq}$ expressed as a sum over "Z-avoiding" Delannoy paths in a $p \times q$ grid. An efficient dynamic-programming implementation of this formula computes the volume in dimensions far beyond the reach of general-purpose exact-volume algorithms (e.g., the volume of $\mathcal{M}_{20,20}$, which has dimension 399, is computed in a fraction of a second). As a corollary of our Stanley decomposition, we also obtain a combinatorial closed form for the Ehrhart series of $\mathcal{M}_{pq}$.

For a graph $\Gamma=(V,E)$ and nonnegative integers $a$ and $b$, a nonempty proper subset $C \subset V$ is called an $(a,b)$-regular set if every vertex in $C$ has exactly $a$ neighbors in $C$, and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In this paper, we study the existence of such sets in connected Cayley graph $\Gamma = \operatorname{Cay}(\mathbb{Z}_n, S)$. We establish a necessary and sufficient condition for the existence of $(0, |S|)$-regular sets and identify additional conditions under which no such set can exist. We further prove that $(|S|, 0)$-regular sets do not occur in $\Gamma$, and more generally, that no connected Cayley graph $\operatorname{Cay}(G,S)$ contains a $(1, |S|)$-regular set. As a main result, we determine the existence and nonexistence of $(a,b)$-regular sets in connected circulant quartic graphs for all possible values of $a$ and $b$.

A graph is locally irregular if adjacent vertices have distinct degrees. A graph G is decomposable if its edge set can be decomposed into locally irregular graphs, and its locally irregular chromatic index lir(G) is the least number of graphs in such a decomposition. We prove that lir(G) <= 18 for every decomposable graph G, improving the previous bound of 220.

The betweenness centrality of a vertex $v$ in a graph $G = (V,E)$ is the sum of the relative numbers of shortest paths of $G$ that pass through $v$. The vertices of $G$ which have the maximum (resp. minimum) betweenness induce the betweenness center (resp. betweenness periphery) of $G$. We study betweenness of graphs and their localization in graph blocks, presenting sufficient conditions for graphs (in terms of diameter or block sizes) to have those centers contained in a single block. Further, we show that each graph occurs as the subgraph induced by the betweenness center of some graph (as well as the subgraph induced by the betweenness periphery). For trees, we show, by an alternative proof, that their betweenness center is always contained in a path; in addition, we enumerate trees of order at most 20 according to the order of their betweenness centers.

The subset $A = \{a_i:i \in I\}$ of a normed vector space is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. Let $\varepsilon = \{\varepsilon_i:i \in I\}$ be a set of positive real numbers. An $\varepsilon$-perturbation of $A$ is a set $A' = \{a'_i:i\in I\}$ such that $|a_i'-a_i|<\varepsilon_i$ for all $i \in I$. Let $\Delta_{hA} = \inf\{|x'-x| : x,x' \in hA \text{ and } x\neq x'\}$. It is proved that if $A$ is finite or countably infinite set with $\Delta_{hA}>0$, then there is a $B_h$-set $A'$ that is an $\varepsilon$-perturbation of $A$.

As a generalisation of the Erdős-Sós conjecture about graphs, Addario-Berry, Havet, Linhares Sales, Reed and Thomassé conjectured that every digraph on $n$ vertices with more than $(k-1)n$ arcs contains every antidirected tree with $k$ arcs. We prove a dense, approximate version of this for trees with bounded maximum degree, as well as for trees whose layers are evenly distributed. We use a regularity based approach, centred around finding a copy of a given tree in the blow up of a caterpillar.

In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus $g$ and degree $d = g/2 + 1$, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number $C_{g/2} = \frac{1}{g/2+1}\binom{g}{g/2}$. In this paper, we construct the tropical counterpart of this universal family: a polyhedral cone complex $\mathcal{G}_{g \to 0, d}^{\mathrm{trop}}$ parametrizing degree-$d$ tropical morphisms from genus-$g$ metric graphs to metric trees. For even $g$ and $d = g/2 + 1$, we prove that the forgetful projection $\Pi \colon \mathcal{G}_{g \to 0, d}^{\mathrm{trop}} \to \mathcal{M}_{g}^{\mathrm{trop}}$ is a branched cover of degree $C_{g/2}$ equipped with natural determinantal multiplicities. We compute this degree by showing that on caterpillars of loops the morphisms are in bijection with ballot sequences, and we establish its global invariance across $\mathcal{M}_{g}^{\mathrm{trop}}$ via a tropical balancing condition across codimension-$1$ walls. Via deformation and path lifting, this yields an effective method to construct Catalan-many gonality-witnessing maps for any generic metric graph, establishing that the tree gonality of any genus-$g$ metric graph is at most $\lceil g/2 \rceil + 1$.

Fang and Huang proved that every cancellative pair $(\mathcal{A},\mathcal{B})$ of families of subsets of $[n]$ satisfies $|\mathcal{A}|\cdot|\mathcal{B}|\leq(\frac{9}{4})^n$. Their proof uses entropy. We give a Fourier-analytic proof based on Lifshitz's one-sided noise operator $\mathrm{T}^{1/4\to1/2}$. The main tool is a near-$L^1$ hypercontractive estimate for this operator.

We give several characterizations of an inhomogeneous basis of the ring of symmetric functions whose evaluations are the character values of the irreducible representations of the rook monoid (symmetric inverse semigroup). Using Schur–Weyl duality, we show that the transition coefficients of this basis with the power symmetric basis are the characters of the propagating partition algebra (dual symmetric inverse monoid algebra). In addition, the structure coefficients of this basis are equal to the coefficients in the smash (or Heisenberg) product of Schur functions.

We study simultaneous color densities in packing colorings of integer distance graphs. For $D(1,6)$, we determine several exact density regions and prove that colors $1$ through $7$ have maximum combined density $211/252$. When this maximum is approached, the seven individual color frequencies are forced to converge to a specified vector. On an optimal low-color layer, some density vectors have nonperiodic realizations but no periodic realization; we determine how much accumulated density loss is necessary for switching between the relevant configurations. For sufficiently large additional color indices, a fixed finite graph describes the joint density region. In particular, we determine a seven-vertex region for every $i\equiv8\pmod{14}$ with $i\ge36$ and prove that $36$ is the first stable index in this residue class. The proofs combine finite-state integer certificates with explicit constructions and limit arguments. Applications give $17\le\chi_\rho(D(1,6))\le20$, $18\le\chi_\rho(D(1,8))\le22$, and $\chi_\rho(D(1,9))\le17$.

A family of paths strongly separates the edges of a graph if every two distinct edges are separated in both directions by paths in the family. Bonamy, Botler, Dross, Naia, and Skokan proved that every $n$-vertex graph admits such a family of at most $19n$ paths. We improve this bound to $10n-o(n)$.

We study $p$-spread probability measures on the Boolean lattice. We show that if a family of sets $A$ is large under the product Bernoulli-$p$ measure, then no $p$-spread measure can be supported on sets that are not covered by the union of two members of $A$, answering the fractional version of Talagrand's discrete convexity problem. Consequently, we establish a coupling theorem between $p$-spread and product Bernoulli-$p$ measures.