Papers

The $\ell$-reconfiguration graph of a graph $G$, denoted by $\mathcal{R}_{\ell}(G)$, is the graph whose vertices are the proper $\ell$-colorings of $G$, with an edge between two colorings if they differ in color on exactly one vertex. For any graph $G$ of treewidth at most $2$, Bousquet and Perarnau showed that $\mathcal{R}_\ell(G)$ has linear diameter for $\ell\geq 6$. This result was later extended by Bartier, Bousquet, and Heinrich, who proved that $\mathcal{R}_5(G)$ also has linear diameter. In this paper, we show that for each $\ell\geq 5$, the $\ell$-reconfiguration graphs of $K_{2,3}$-minor-free graphs, some of which include graphs of treewidth $3$, have linear diameter. As a key step in our proof, we also establish that the $(\ell-1)$-reconfiguration graphs of cactus graphs have linear diameter.

A linear code is referred to as a locally repairable code (LRC) with locality r if any erased code symbol can be recovered by accessing at most r other code symbols. LRCs are highly desirable for distributed storage systems to enhance repair efficiency. In this paper, we investigate LRCs with disjoint repair sets via the parity-check matrix method. Firstly, we propose a novel concept of the s-Pasch configuration and present a geometric characterization for the existence of LRCs with minimum distance 5 and locality 3. Subsequently, we construct k-optimal LRCs by exploiting the point-line relationship in PG(2,q). Finally, a family of q-ary k-optimal LRCs with minimum distance 6 and general locality r is constructed using partial r-spreads.

This paper studies sum-of-squares (SoS) representations of nonnegative polynomials over the hypercube $[0,1]^n$. De Klerk and Laurent (SIAM J. Optim., 2010) conjectured that the smallest constant $C_n$ such that the polynomial $x_1\cdots x_n +C_n$ is contained in the degree-$n$ truncated quadratic module $M_{n,n}(x_1-x_1^2,\ldots,x_n-x_n^2)$ of the hypercube is $C_n=1/(n(n+2))$, for $n$ even. We specialize symmetry reduction techniques for finding sum-of-squares certificates to the hypercube, where the generators $x_i-x_i^2$ are not individually invariant under the symmetric group but form an invariant set, and apply them to this conjecture. Combining this reduction with a further (heuristic) sparsity reduction, a rational rounding step, and an exact verification over $\mathbb{Q}$, we prove the bound $C_8\leq 11/1000 <1/80$. In particular, this disproves the conjectured optimal value for $n=8$.

In this paper, we investigate the notion of the \textit{$\lambda$-chromatic polynomial} of a graph, which enumerates the number of distinct $L(2,1)$-colorings using colors from a prescribed finite set. We prove that the $\lambda$-chromatic polynomial of a graph with $n$ vertices is a monic polynomial of degree $n$ and provide a combinatorial interpretation via lattice point enumeration within the framework of inside-out polytopes. Moreover, we compute the $\lambda$-chromatic polynomial of complete graphs $K_n$ using lattice path enumeration, and we develop a block-gap technique to derive the $\lambda$-chromatic polynomials for complete bipartite and multipartite graphs. Our approach unifies geometric, combinatorial, and algebraic methods to provide a systematic treatment of $\lambda$-colorings across various families of graphs.

Let $(x_n)_{n\geq 1}$ be a sequence of distinct points on the unit circle. After the first $n$ points are inserted, the circle is divided into $n$ intervals. For a fixed integer $r\geq 1$, let $M_n^{(r)}$ and $m_n^{(r)}$ denote respectively the largest and smallest total lengths of $r$ consecutive intervals. A theorem of de Bruijn and Erdős gives \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\geq 1+\frac1r . \] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\geq 2$ remain much less understood. We prove the improved lower bound \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{r}{r^2-1} \qquad (r\geq 2). \] In particular, for two consecutive intervals the lower bound becomes $5/3$, improving the de Bruijn–Erdős bound $3/2$.

We introduce a diffusion-based uncertainty model for robust optimization on directed graphs, in which perturbations of edge weights propagate along adjacent edges and satisfy conservation constraints at nodes. This topology-aware structure is natural in networked systems where uncertainty is induced by flows and local interactions, including transportation, logistics, communication, and energy networks. We analyze how such diffusive uncertainty reshapes the computational landscape of robust graph optimization. We focus on two canonical combinatorial graph problems, shortest path and the traveling salesman problem (TSP), which provide complementary benchmarks: shortest path is polynomial-time solvable in the nominal setting, whereas TSP is already NP-hard. We show that, for shortest path, propagation depth induces a sharp transition between tractable and intractable robust counterparts. For the traveling salesman problem, robustness often adds no computational complexity beyond ordinary TSP, because the structure of Hamiltonian cycles makes the fixed-tour adversarial problem collapse to explicit formulas. Together, these results show that topology-aware uncertainty can fundamentally change robust combinatorial optimization, with tractability governed by the interaction between propagation, budget geometry, and the structure of feasible solutions.

Under what conditions does the sign of the Ollivier Ricci curvature on a graph of a certain order change? In this paper, we discuss the curvature of graphs obtained by removing edges from complete graphs, as complete graphs have a stable positive curvature. We defined graphs obtained by removing matching edges, the set of edges incident with the vertex, and cycle edges from complete graphs, and then analyzed the Ollivier Ricci curvature of those graphs. The results show that the curvature of the graphs in the above three patterns is equal to the value obtained by dividing the number of triangles, including two vertices, by the maximum degree of the two vertices. This result also indicates that the curvature of the above graphs is zero or positive. This study concludes that the Ollivier Ricci curvature is predicted to be positive even if some edges are removed from a complete graph, and we suggest that these discussions are suitable for investigating the conditions under which the sign of the Ollivier Ricci curvature on a graph.

We introduce a spectral version of the classical inducibility problem. Given an $\ell$-vertex graph $F$ and an $n$-vertex graph $G$, let $H_F(G)$ be the $\ell$-uniform hypergraph whose edges are the $\ell$-sets inducing a copy of $F$ in $G$. We study the maximum possible $\alpha$-spectral radius of $H_F(G)$ over all $n$-vertex graphs $G$. For fixed $G$, this spectral parameter tends to $\ell!$ times the number of induced copies of $F$ in $G$ as $\alpha\to\infty$, and therefore refines the usual induced-copy count. Our main result is a spectral analogue of the Brown–Sidorenko reduction: for every complete multipartite graph $F$, every $n$, and every $\alpha\ge1$, a spectral extremal graph can be chosen to be complete multipartite. We also show that the leading asymptotic constant is the ordinary inducibility $i(F)$, and obtain exact multipartite reductions for stars $K_{1,t}$ and balanced complete $r$-partite graphs $K_{a,\ldots,a}$ with $r\le 2^a-1$.

We develop a systematic method to express generating functions for moments of combinatorial statistics in terms of partition traces. We employ an algebraic approach based on the complete Bell polynomials and their inversion formula, alongside an analytic approach via Faà di Bruno's formula. Our approach can be applied to a wide class of combinatorial statistics, such as the largest part of an integer partition, the partition crank and rank, and the unimodal sequence rank.

The Moebius-Kantor graph MK=G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order 16. In topological graph theory, it illustrates the Heawood number 7 of the torus and leads to the Tucker group Aut(MK), the unique group of genus 2. We compute the Lefschetz numbers to illustrate the Brouwer-Lefschetz fixed point theorem. MK is also the dual of the 2-skeleton complex of the 3-sphere G. The graph represents one of flat Clifford tori of a Hopf fibration in the 3-sphere G=K(2,2,2,2) reflecting that Coxeter saw that MK is a subgraph of the tesseract G*. It carries a metric d so that (MK,d) has only one algebraic group structure (P(1),*) that preserves the metric. It makes the Pauli group natural, similarly as the Moebius ladder M(16) makes the dihedral group D(16) natural, forcing the algebraic structure from the metric structure.

We prove, at arbitrary loop order and including arbitrary tadpole decorations, the shifted-tree decomposition of two-site cosmological wavefunction coefficients proposed in our previous work, and uncover the combinatorial and geometric structure underlying it. For conformally coupled scalars with polynomial interactions, this decomposition expresses each loop coefficient as a finite signed sum of tree coefficients evaluated at shifted energies. For bubble-like graphs, the alternating sum over a Boolean lattice can be written in terms of finite differences or maximal chains. We prove the maximal-chain identity algebraically by induction on loop order and geometrically by decomposing a cubical Laplace integral into simplices. The maximal chains correspond to permutohedron vertices, whose incident facets encode the chain-dependent divisors. For tadpole-decorated graphs, the finite-difference Leibniz rule gives an explicit expansion indexed by ordered partitions and local maximal chains, which glue into global maximal chains and organize the coefficient as a combinatorial "path integral". A natural ordering of the divisors yields a term-by-term match with graph tubings and completes the all-loop proof, while other orderings give alternative tubing representations of the same coefficient. These identities hold before spatial loop-momentum integration and extend to power-law FRW backgrounds by linearity of the twisted integral. Two-site loop coefficients can therefore be obtained directly from known tree coefficients without solving new loop-level differential systems.

Let $\bigtriangleup$ be a simplicial complex and let $\delta_{\mathcal{NF}}$ denote the NF-operator. The NF-complex $\delta_{\mathcal{NF}}(\bigtriangleup)$ is defined as the Stanley–Reisner complex of the facet ideal of $\bigtriangleup$. Iterating $\delta_{\mathcal{NF}}$ gives a periodic orbit (up to isomorphism), and the smallest positive integer $t$ for which $\delta_{\mathcal{NF}}^{\,t}(\bigtriangleup)\cong \bigtriangleup$ is called the NF-number of $\bigtriangleup$ (Habi and Mahmood, Algebra Colloquium, 2022). In this work, we provide various results and determine explicit formulas for the NF-number for several families of graphs. In particular, we compute the NF-number for dumbbell graphs. We also prove that the NF-number of the complete split graph $S_{n,m}$ equals $m+n+2$, and that the NF-number of the double star $D_{p+q}$ equals $p+q+4$. We conclude with remarks, open problems, and conjectures to guide future research.

We define a quantum graph to be vertex-transitive if the join of its automorphism group is the maximum quantum relation on its quantum vertex set, in direct analogy with the classical case. All simple quantum graphs in $M_2(\mathbb C)$ are vertex-transitive, but many simple quantum graphs in $M_3(\mathbb C)$ are not vertex-transitive. We provide a complete classification of vertex-transitive quantum graphs in $M_3(\mathbb C)$ up to isomorphism. To do this, we introduce a polynomial invariant for quantum graphs in $M_n(\mathbb C)$, which we call the panoramic polynomial.

A distance-k dominating set of a graph is a set of vertices such that every vertex lies within distance k of some vertex in the set; its minimum size is the distance-k domination number gamma_k. We study gamma_k for triangular matchstick graphs T_d. Harris et al. (2020) claimed two upper bounds for the cases k=1 and k=2. These bounds are shown to be incorrect and are replaced by a corrected general upper bound for gamma_k(T_d) obtained by refining their tiling method. For k=1 and d=7q+beta, where 0 <= beta <= 6, we further sharpen this construction and prove gamma(T_{7q+beta}) <= 3.5q^2+(beta+4.5)q+(beta-1). Finally, we determine the radius of T_d as r_d = ceil(2d/3), which implies gamma_k(T_d)=1 whenever k >= r_d.

Rational functions make sense over any commutative ring, as long as the denominators are invertible. When there is no subtraction involved, they even apply over semirings (rings without subtraction). It is particularly worthwhile to evaluate them over the *tropical semiring*, in which the roles of addition and multiplication are played by maxima and addition. Over this semiring, algebraic results often acquire combinatorial meaning. We give a few examples.

Killing forms on finite groups arise as special cases of braided Killing forms on braided Lie algebras. If $\mathcal{C}$ is a conjugation-stable subset of a finite group $G$, the Killing form on $\mathbb{C}\mathcal{C}$ is given by $K_\mathcal{C}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b \in \mathcal{C}$. It is conjectured in previous work by López Peña, Majid and Rietsch that $K_\mathcal{C}$ is non-degenerate for any real conjugacy class $\mathcal{C}$ in a finite simple group. In this article, we reformulate the conjecture and introduce combinatorial conditions - the $\textit{1-element condition}$ and the $\textit{2-element condition}$ - that are sufficient for non-degeneracy to hold. This allows us to prove the conjecture for simple groups of the form $\mathrm{PSL}_2(q)$ and certain conjugacy classes in the alternating and symmetric groups. Moreover, we verify computationally that every real conjugacy class in a simple group of order $\leq 10^9$ fulfills at least one of these two conditions, thereby significantly extending the computational evidence for the conjecture. This raises the question whether these conditions are satisfied by all conjugacy classes in finite simple groups.

For a simple graph $\Gamma$ and a real number $\lambda$, the general reduced second Zagreb index is defined by the formula $$GRM_\lambda(\Gamma)=\sum_{ab\in E(\Gamma)}[(\deg_{\Gamma}(a)+\lambda)(\deg_{\Gamma}(b)+\lambda)]\,.$$ A sharp lower bound for $GRM_\lambda$ over all trees of given order and maximum degree under the condition that $\lambda\ge -\frac{1}{2}$ is established. A parallel result is proved for unicyclic graphs under the condition $\lambda\ge -\frac{1}{2}$. The corresponding minimal trees and unicyclic graphs are identified. These findings improve upon the lower bounds previously established by Buyantogtokh, Horoldagva, and Das concerning $GRM_\lambda$ of trees and unicyclic graphs of given order.

Brualdi, Friedland, and Pothen studied sparse bases of row spaces and proposed a rank-intersection criterion for elementary row-space vectors of sparse generic matrices. We give a \(4\times 8\) sparse generic counterexample. The example has four elementary vectors in its row space whose zero sets satisfy all the proposed rank-intersection inequalities, but the four vectors satisfy a nontrivial linear relation and hence do not form a basis. The construction also explains the obstruction: the proposed inequalities see only ranks of column sets indexed by intersections of zero sets, while linear independence of the corresponding elementary vectors is governed by the relative position of the hyperplanes annihilated by their coefficient vectors.

Modular Invariant Theory is a branch of mathematics that explores the behavior of polynomial functions invariant under group actions, particularly over fields with positive characteristic. Overall, modular invariant theory serves as a vital link connecting algebraic methods with combinatorial and topological applications. Based on the existing literature, this undergraduate thesis aims to investigate conjectures and problems emerging from the work of Macdonald (1992), and the recent work of Lewis, Reiner, and Stanton (2017), as well as subsequent developments by L. M. Ha, N. D. H. Hai, and N. V. Nghia (2024). In particular, we state and prove a generalization of Conjecture (7.25) by Macdonald (1992), leading to an extension of the Stong-Tamagawa formula, which is a basis-free characterization of Schur functions over finite fields. Besides, we examine conjectures by Lewis-Reiner-Stanton (2017) about invariant spaces of truncated polynomial algebras under the action of parabolic subgroups, and the proof for the Borel subgroups by Ha, Hai, and Nghia (2024), from which we extend the investigation to the invariant rings under the unipotent group's action. Additionally, we consider the delta operators-a pivotal family of operators used in the proof of Ha, Hai, and Nghia (2024)-with particular attention to their polynomiality properties.

In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on $(n,k,L)$-systems. The main result is a proof of the Aljohani–Bamberg–Cameron conjecture which claims that if $n > n_0(k)$ and there are an $(n,k,L)$-system and an $(n,k,\{0,\dots,k-1\}\setminus L)$-system whose sizes have product $\binom{n}{k}$, then they are a $t$-intersecting family and a Steiner system $S(t,k,n)$ for some $t$.

We review known linear and matrix generalizations of Hall's classic “marriage theorem” and Kőnig's theorem on partial matchings in bipartite graphs, and relate them to linear and matrix generalizations of Dilworth's theorem about chains and antichains in posets and Menger's theorem about disjoint paths in directed graphs.

For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles of area $1/2$. We prove that the measures of such sets satisfy $|A|= O_c(R^2/(\log R)^c)$ for any constant $c<1/4$. An ingredient in the proof is a hyperbolic variant of the two-dimensional trilinear smoothing inequality by Christ, Durcik, and Roos. The aforementioned upper bound is complemented with an example of a set of measure $\Omega(R\log R)$ avoiding the same point configuration. Next, we study measurable sets $A\subseteq [0,R]^2$ that avoid triples of points spanning a triangle of a given fixed area and establish a sharpening of the aforementioned upper bound to any $c<1/2$. This makes partial progress on a question by Erdős, who conjectured an upper bound $O(1)$, and improves over a quantitatively weak $o(R^2)$ result by Graham. The latter proof additionally uses induction on scales to interchangeably control the density and the Riesz energy of the set $A$.

We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this new mex statistic for several classical combinatorial structures, by providing the mex generating function and/or a closed formula for its coefficients in each of the cases.

We consider Subtraction Nim, where two players have exactly the same options, but which is partizan in the sense that at the game ending, a partizan rule is applied for the decision of the winner. We consider the following example: Let $S$ be the set of removable numbers, which is a non-empty finite subset of positive integers greater than or equal to $2$, applied for both players Left and Right. At the end of the game, Left wins if the number of remaining tokens is even, and Right wins if the number of remaining tokens is odd. We computed the outcomes for many $S$, and found surprising phenomena that in most examples of $S$ (almost $98\%$ of some samples), the outcomes are $\mathcal{L}$-positions for all large enough $n$. In comparison, $\mathcal{R}$-positions appear only occasionally. The main theorem explains why this phenomenon occurs. We prove that $n+1$ and $n-1$ are $\mathcal{L}$-positions when $n$ is an $\mathcal{R}$-position. Similarly, $\mathcal{L}$-positions appear whenever $\mathcal{P}$-positions or $\mathcal{N}$-positions appear. Only $\mathcal{L}$-positions can last forever.

We study a question of Harju from 2019 regarding the existence of infinite ternary square-free words whose subsequences modulo $p$ and $q$ are also square-free for relatively prime integers $p$ and $q$. Among such pairs $(p, q)$ with $p, q \geq 3$, the only two pairs with this property known prior to this work were $(3, 11)$ and $(5, 6)$. We prove that there are finitely many pairs $(p, q)$ of relatively prime integers with $p, q \geq 3$ for which there is no infinite ternary square-free word whose subsequences modulo $p$ and $q$ are square-free. To prove our result, we combine different techniques, including the construction of words from multi-valued square-free morphisms and circular square-free morphisms. We also introduce the notion of square-free transducers, a generalization of square-free morphisms that may be of independent interest.

We introduce variants of the Maker-Breaker and Waiter-Client games, which we call stotting, in which a player grants a slight advantage to the opponent. We prove that a winning strategy in either stotting variant yields winning strategies for both Maker and Waiter in the classical setting. Several existing Maker strategies in the literature in fact win with stotting, and therefore automatically provide both classical winning strategies (and similarly for stotting Waiter). Knox previously disproved a conjecture of Beck asserting that whenever Maker wins the Maker-Breaker game, Waiter also wins the corresponding Waiter-Client game; in this sense, our framework may be viewed as a way of repairing Beck's conjecture.

Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. As an application, we show that a uniformly at random chosen subset $A \subseteq \{1, \ldots, n\}$ of size $k$ such that $|A + A| \le \lambda k$, for $\lambda = o(k^{1/4} / \log^{1/2} n)$, with high probability belongs to an arithmetic progression of length $\lambda k / 2 + o(\lambda k)$. Both proofs are based on a bespoke container argument, exploiting the additive robustness of the considered sets.

Let $G_1$ denote the incidence graph of the complete graph $K_{q+1}$. We study limited augmented Zarankiewicz numbers in this family through structured 0–1 optimization models for admissible 2-edge augmentation. For the original framework, we combine exact ILP formulations for the smallest instances with constructive search followed by exact admissibility verification for larger instances. This yields \[ {z_L(6,4)=14,\qquad z_L(10,5)=26,\qquad z_L(15,6)\ge 43,\qquad z_L(21,7)\ge 64,\qquad z_L(28,8)\ge 88.} \] The first two values are exact, whereas the latter three are rigorous lower bounds obtained from explicitly verified admissible families. We then formulate the global weak problem determined by \((S),(W2),(W2'),(W3)\). On the larger incidence-family cases, the corresponding global exact optimization problem is currently computationally out of reach, and our computations provide the certified lower bounds \[ {z_{WL}(6,4)=14,\quad z_{WL}(10,5)\ge 29,\quad z_{WL}(15,6)\ge 46,\quad z_{WL}(21,7)\ge 65,\quad z_{WL}(28,8)\ge 90.} \] On the fourteen literal-grid benchmarks from $(4,3)$ through $(7,7)$, by contrast, the global weak problem is solved exactly. Across this exact benchmark block, the weak framework is never worse than the strong framework and is strictly better in several cases. These computations improve the corresponding classical Zarankiewicz numbers and therefore strengthen available lower bounds for $\operatorname{BSR}(m,n)$ within this family.

We derive the local and central limit theorems for the Stirling numbers of the second kind by elementary means, obtaining as corollaries effective asymptotic estimates for the Bell numbers and for the moments of the distribution. We also develop asymptotic expansions along several directions, all based on a novel finite-differencing approach; this provides the first self-contained elementary justification of such expansions.

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\alpha=\alpha(n)=o(1)$, let $G_{\alpha}$ be an $n$-vertex graph with minimum degree $\delta(G_{\alpha})\ge\alpha n$. We prove that if $$p\ge(1+\varepsilon)\frac{\log(1/\alpha)}{n},$$ then the union $G_{\alpha}\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $\alpha n\rightarrow\infty$, thereby establishing the exact probability threshold for Hamiltonicity in this sparse regime. Our proof relies on a robust random expansion lemma, Pósa's booster lemma, and a sprinkling argument.