Papers

A long-standing question is whether chromatic symmetric functions can distinguish non-isomorphic trees. Campbell introduced chromatic noncommutative symmetric functions for digraphs, which lift chromatic symmetric functions to NSym, and asked to what extent they can distinguish non-isomorphic oriented trees. In this article, we prove that chromatic noncommutative symmetric functions can reconstruct oriented stars, oriented double stars, some oriented caterpillars, and some oriented paths.

The 2026 disproof of Erdős's unit-distance conjecture and Sawin's quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $\delta=0.014114\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $\delta=0.015263\ldots$ and supporting the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. For extended ramified prime ranges, the Emmerich–Cordella certificate obtained with the same framework reports $u(n)>n^{1.031}$ for $\#T=67$, illustrating the importance of enlarging $T$. Very recent MathOverflow discussions, brought to the author's attention as of version~4, report further improvements, including certificates above $\delta>0.035$ and beyond $\delta>0.036$. Some of these improvements may rely not only on larger prime ranges but also on modified constraint systems and additional degrees of freedom that deviate from Sawin's original formulation. Beyond this application, the work illustrates how randomized optimization heuristics can improve, verify, and refine explicit certificates for combinatorial geometry through nonlinear integer optimization.

We present a framework to generalize the standard b-ary Gray code to get the k-bonacci ones obtained in [5] as well as many others by using theoretical tools that allow to make calculations on lists. We introduce the notion of Z-Gray product, from which we deduce sequences of lists of finite words avoiding a predefinite list Z of factors and which satisfy a power-associativity property as well a generalizations of the classical flipping digit property.

We prove an upper bound for the number of subgroups of an abelian $p$-group, with constants that are sharp on homocyclic blocks. If $A_\lambda$ has type $\lambda$, the main exponent is $\eta(\lambda)=\sum_i\lfloor(\lambda'_i)^2/4\rfloor$, where the parts of $\lambda'$ are the column heights of the Ferrers diagram of $\lambda$. The leading coefficient is the number of central choices in the blocks of columns of odd height. The proof starts from Delsarte's formula, separates the blocks by column height, and estimates each homocyclic block by successive Durfee squares. We also give three applications: a comparison theorem for $p$-groups of nilpotency class less than $p$, diagonal summatory estimates with the shape fixed, and formulae for leading terms at fixed rank for finite abelian $p$-groups.

Johnson graph are a family of graphs that play an important role in the theory of constant-weight codes, extremal combinatorics, and combinatorial geometry. We study signed analogues of classical Johnson graphs, denoted by $J_\pm(n,k)$, whose vertices are vectors of the form $\pm e_{i_1}\pm\cdots\pm e_{i_k}$, where two vertices are adjacent whenever their dot product equals $k-1$. We are particularly interested in maximum independent sets in the case $k=4$. An example of such an independent set in $J_\pm(n,4)$, which we call classical, is obtained by lifting an arbitrary optimal $(n,4,4)$-code. Such independent sets naturally define kissing arrangements in ${\mathbb R}^n$. We develop an algorithm that is practical for computing all maximum independent sets in $J_\pm(n,4)$ up to signed permutations for $n\le 12$, $n\ne 11$. In addition to obtaining complete lists, we provide structural characterizations of all types of maximum independent sets in these dimensions, excluding $n=5$ and $n=11$. Our most striking results concern the case $n=12$. We identify $1579$ non-isomorphic maximum independent sets in $J_\pm(12,4)$, all corresponding to non-isometric kissing arrangements of size $840$ in ${\mathbb R}^{12}$. Structurally, $1575$ of these independent sets arise from three different constructions, the rest are liftings of one of four $(12,4,4)$-codes. To our knowledge, this is the first dimension in which such a large diversity of potentially optimal kissing arrangements has been observed. Beyond this finite range, we prove that for $n\equiv 2$ or $4 \pmod 6$, every maximum independent set arises from a Steiner quadruple system. We also obtain a characterization of the so-called nontrivially self-compatible codes, namely optimal $(n,4,4)$-codes from which non-classical maximum independent sets can be constructed.

In this paper, we consider triangle-free graphs with diameter 2. If a triangle-free graph $G$ with diameter 2 is not isomorphic to a star, then the radius of $G$ is also 2, where such a graph is also called a $2$-self-centered graph. Shekarriz et al. [A characterization for 2-self-centered graphs, Discuss. Math. Graph Theory 38 (2018), 27–37.] gave a characterization of 2-self-centered graphs. However, there is a slight flaw in their characterization. Thus, in this paper, we modify it and prove an accurate characterization of those graphs. Furthermore, by using our characterization, we prove some results concerning the chromatic number of triangle-free graphs with diameter 2.

Efforts have been made to extend Cassini's identity (also known as Simson's identity) to the k-step or k-bonacci numbers for decades. These efforts have lacked both completeness of result and simplicity of proof, and this question remains open and relevant. In this note, we offer a definitive solution as well as the generalization of both Catalan's and Vajda's identities.

The Membership Problem for Pedigree Polytope (M3P) asks, given $X\in\mathbb{Q}^{\binom{n}{3}}$, whether $X\in\mathrm{conv}(P_n)$, where $P_n$ is the set of all pedigrees. A pedigree is a structured encoding of a Hamiltonian cycle construction in $K_n$. We establish that M3P is solvable in strongly polynomial time via a recursively constructed layered network $(N_k, R_k, \mu)$ and a multicommodity flow problem MCF$(k)$. The necessary and sufficient condition for membership established is that the optimal total flow in MCF$(n-1)$ equals the maximum possible flow $z_{\max}$. The complexity analysis, grounded in Tardos's strongly polynomial algorithm for combinatorial linear programs (1986), shows that this condition can be checked in strongly polynomial time in the dimension of the matrix involved. By sufficiency, this implies M3P~$\in$~P. Since the Symmetric Travelling Salesman Problem (STSP) reduces to M3P via the Multistage Insertion (MI) formulation (Arthanari 1983), STSP is solvable in polynomial time, and the P vs.NP question is resolved. The proofs leading to this result are fully machine-verified in Lean~4/Mathlib4, with zero unresolved sorrys in the main proof chain. The main contribution is the Lean~4 machine verification of all proofs in the main chain, resulting in theorem p\_equals\_np: P = NP. The Lean~4 formal verification covers the sufficiency of MCF(n-1) for membership in $\mathrm{conv}(P_n)$, and the P = NP chain via Maurras (2002), Grötschel–Lovász–Schrijver (1988), Cook (1971), and Karp (1972). The complete lean project (36 Lean~4 files, 2968/2968 build targets clean) is available at https://github.com/TiruArt/Pedigree-Polytopes-Lean4.

The partial Petrial polynomial of a ribbon graph, introduced by Gross, Mansour and Tucker, enumerates partial Petrials by Euler genus. Recently, Deng, Jin and Yan defined an analogue for grafts and showed that it can be expressed as a rank-generating function of an adjacency matrix. In this paper we first prove a recurrence relation that reduces the partial Petrial polynomial of a graph with respect to an arbitrary edge, expressing it as a sum of three terms involving graphs obtained by local complementation and edge pivoting. This recurrence extends the known leaf-reduction formula to vertices of any positive degree. Second, using this recurrence we compare the lowest and highest degree coefficients of the polynomial. We prove that the lowest coefficient is always at most the highest coefficient, and that equality holds if and only if the graph has no edges.

We prove a conjecture of Glasby, Praeger, and Unger concerning the symmetric group $S_{n}$. Let $\pi_{n}$ denote the proportion of elements of $S_{n}$ that are pre-$p$-cycles for some prime $p\in[2, n-3]$. We prove that $\pi_{n} > 1/3$ for all $n\geq 8$.

While the uncrowding algorithm on set-valued tableaux has long been instrumental in proving the Schur positivity of stable symmetric Grothendieck polynomials, lattice models have emerged as a modern framework for investigating symmetric functions, in particular symmetric Grothendieck polynomials. In this work, we synthesize these combinatorial and lattice-theoretic approaches by defining both the Robinson–Schensted–Knuth (RSK) correspondence and the uncrowding operation directly on a 5-vertex model of Motegi and Sakai and its subsequent reinterpretation by Buciumas, Scrimshaw, and Weber. Our lattice-based RSK formulation yields a powerful new result: the direct construction of the associated crystal structure on the states of the 5-vertex model.

The Distance Geometry Problem (DGP) asks for a geometric realization of a weighted graph with \(n\) vertices in \(\mathbb{R}^K\) such that Euclidean distances between vertices match the given edge weights. When a vertex order where every non-seed vertex has \(K\) predecessors inducing a clique is part of the input, the search space can be discretized via \(K\)-lateration, branching into a binary tree. In this subclass, known as the Combinatorial Discretizable Distance Geometry Problem (Combinatorial DDGP), the goal is to determine the number of realizations satisfying all distance constraints. While the number of realizations is almost always \(2^{n-K}\) when no additional distance constraints are present, a topological solution count in the presence of additional constraints has remained elusive. We develop an algebraic rank-count theory for the feasible binary branch codes, proving that under mirror-separated parameters they form an affine space over \(\mathbb{F}_2\) whenever a viable reference solution exists.

We prove a sparse transference theorem for induced Ramsey graphs. The theorem transfers the weighted random-host proof of Aragão, Campos, Dahia, Filipe, and Marciano to the sparse random setting. It follows that, for every fixed graph $H$ with no isolated vertices and at least two edges, and every $\eta>0$, there is $C>0$ such that, whenever $N\ge r^{Cr}$ and $N^{-1/m_2(H)+\eta}\le p\le \frac12$, with high probability every $r$-colouring of the edges of $G(N,p)$ contains a monochromatic induced copy of $H$. Here, $m_2(H)$ denotes the usual maximum 2-density of $H$.

The maximal entries of Dynkin friezes over the positive integers have recently been determined for all finite Dynkin types except $B_n$ and $D_n$. In this note, we explicitly construct large positive integral points on affine cluster varieties of type $B_n$ (resp. $D_n$), giving rise to friezes of types $B_n$ (resp. $D_n$) over the positive integers with largest entries $F_{n+1} F_{n+2} - 1$ (resp. $F_n F_{n+1} - 1$) where $F_k$ is the $k$-th Fibonacci number. We conjecture that these are the maximal possible entries for their respective Dynkin types.

We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length $\Theta (\log \log n)$, it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent $\tau \in (2,3)$ and geometric decay parameter $\alpha > \tau - 1$, geometric routing succeeds with constant probability and finds ultra-short paths of length $\Theta (\log \log n)$, matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network's geometry alone implicitly guides the path to the target through its high-weight core.

The generalized Turán number $ex(n, K_{t, t}, K_{2, t+1})$ is the maximum number of copies of $K_{t, t}$ that a $K_{2, t+1}$-free graph on $n$ vertices can contain. Recently, Pohoata, Tidor, and Yu established that $ex(n, K_{t, t}, K_{2, t+1}) = \Theta_t(n^2)$ for all integers $t \geq 3$. In this short note, we use an explicit construction to establish that when $t$ is a prime power and $n = t^{2e - 1}$, then $$ ex(n, K_{t, t}, K_{2, t+1}) = (1 + o(1))\frac{n^2}{2t(t-1)}. $$

The classical no-$k$-in-line problem asks for the largest number of points that can be placed on an $n \times n$ grid without having $k$ of them collinear. A natural extension, motivated by the analogous question by Erde for $k\in \mathbb{Z}$, is the \emph{extensible no-$(k(n)+1)$-in-line problem}, which seeks a subset of points in $\mathbb{Z}^2$ with maximal possible density such that at most $k(n)$ points are collinear within the subgrid $[1,n]^2$. We construct optimal sets for linear functions and positive-density sets for power functions. We prove that any configuration achieving $\liminf\frac{S_n}{n k(n)} \ge 0.897$ must satisfy $k(n) = \Omega( n^c)$ for some $c>0$ constant; therefore, the extensible no-$k$-in-line problem has no configuration with this property when $k$ is a constant. Finally, we reduce the problem to the extensible no-$k$-in-line problem, showing that if a positive-density point-set exists for a constant limiter function, then one also exists for any sufficiently regular function $k(n)$.

Devnani and Eyyunni recently studied the maps pre$_k$ on integer partitions, which arise from applying elementary symmetric polynomials to the parts of a partition. They asked whether there exists $n \ge 1$ such that exactly one partition of $n$ lies in the image of pre$_2$. We show that this occurs only for $n$ in {1, 2, 4}, and that for all $n \ge 5$, at least two partitions of n are in the image of pre$_2$.

Let $f(k,s)$ denote the minimum integer $m$ such that any family $\mathcal{F}$ consisting of $k$-sized sets of cardinality at least $m$ always contain a sunflower of size $s$. The Erdős-Rado Sunflower Conjecture states that for every $s >2$, there is an constant $C=C(s)$ such that $f(k,s) \leq C^k$. In this paper, we prove the conjecture for shifted families.

Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\binom n k}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $k=2$, Erdős asked whether there exists a constant $\delta>0$ such that $f_2(n)>\delta$ for all $n$, offering \$100 for a proof and \$25 for a disproof. Recently, Bradač and Christoph answered this question in the negative,, proving that $f_2(n)$ tends to $0$, or equivalently that no $n$-vertex graph contains a positive proportion of all $n$-vertex graphs as unique subgraphs. In this paper we show that the situation is fundamentally different for $k$-uniform hypergraphs with $k\ge3$. In particular, for every fixed integer $k\ge 3$, we prove that $\liminf_{n\to\infty} f_k(n) \ge 2/9$.

For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erdős and Graham on canonical van der Waerden numbers.

We provide a classification of neighborly Murai spheres, which implies that all of them are polytopal. Furthermore, we show that each neighborly $d$-sphere with no more than $d+4$ vertices is combinatorially equivalent to a Murai sphere for any $d\geq 1$.

In this thesis we present the combinatorial game theory of the game Rex+. Rex+ is a variant of the game Hex, played on a four sided board made out of hexagons. Both players take turns placing as many stones of their colour as they would like on the board, with the objective being to force the other player to connect their two sides. We describe a new ordering, and present some preliminary results on it.

In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $\rho$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $\rho$ affords the character \[ \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi^{\sigma}, \] for some irreducible complex character $\chi$ of $G$, where $m_{\mathbb{Q}}(\chi)$ denotes the Schur index of $\chi$ over $\mathbb{Q}$, and conversely, every character of this form is afforded by an irreducible representation of $G$ over $\mathbb{Q}$. We obtain a combinatorial description for the counting of inequivalent irreducible $\mathbb{Q}$-representations of $G$ of each distinct degree. Furthermore, we briefly determine the rational character table of $G$ and present a method for constructing an irreducible rational matrix representation $\rho$ of $G$ affording the character $\Omega(\chi)$, where $\chi$ is an irreducible complex character of $G$ arising from parabolic induction. Finally, using the results on the rational representations of $G$, we derive an explicit combinatorial formula, depending only on $q$, for the Wedderburn decomposition of $\mathbb{Q}G$.

Permuted-basement Macdonald polynomials $E_\alpha^\sigma(\mathbf{x};q,t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q,t)[\mathbf{x}]$ for each fixed $\sigma$. There are combinatorial formulas for them as generating functions over composition-shaped non-attacking fillings. In this extended abstract, we bijectively prove identities for the relationship between $E_\alpha^\sigma$, $E_\alpha^{\sigma s_i}$, $E_{s_i\alpha}^\sigma$, and $E_{s_i\alpha}^{\sigma s_i}$. These identities correspond to two combinatorial operations on non-attacking fillings: (1) swapping adjacent entries in the basement, generalizing a result of Alexandersson (2019), and (2) swapping adjacent parts in the shape, which yields a straightening rule for expanding $E_\alpha^\sigma$ in the polynomials $\{E_{s_i\alpha}^\tau\}_\tau$.

An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr–Erdős conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $\Delta\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $\delta(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $\Delta$. We prove this conjecture in a strong form, showing that it is true even if one removes the extra $\varepsilon n$ term in the size of the host graph.

Burr and Erdős conjectured in 1976 that for all integers $k>\ell\geq 0$ such that $k\mathbb{Z}+\ell$ contains an even integer, every $n$-vertex graph without cycles of length $\ell$ modulo $k$ has at most a linear number of edges in $n$. Bollobás confirmed the conjecture in 1977, and Erdős further asked for the exact extremal number. To the best of our knowledge, this problem has been solved only for all residues when $k\leq 4$, and for $\ell\in \{0,2\}$ when $k\geq 5$ is odd. In particular, Bai et al. [arXiv:2503.03504] proved that if $G$ is an $n$-vertex graph with no cycles of length $1$ modulo $3$, then $e(G)\le \frac{5}{3}(n-1)$, and when $9\mid (n-1)$ the equality holds if and only if each block of $G$ is isomorphic to the Petersen graph. Note that for $n> 18$ every extremal graph contains a cut-vertex. In this paper, we investigate the 2-connected setting and determine the maximum number of edges in a 2-connected graph with no cycles of length $1$ modulo $3$. Our results provide a sharp extremal bound and a complete characterization of the extremal graphs, revealing structural differences from the general case. Combining this with the result of Bai et al., we also obtain a complete characterization of all extremal graphs in the general setting, including the cases where $9\nmid (n-1)$. Finally, we determine the maximum number of edges in a $2$-connected graph with no cycles of length $2$ modulo $4$, whose extremal graphs differ substantially from those in the general setting. Consequently, the extremal numbers for $2$-connected graphs with no cycle of a fixed length modulo $k$ are now determined for all $k\leq 4$.

Paired-comparison data are naturally represented by tournaments, where transitivity corresponds to the existence of a global ranking consistent with all pairwise outcomes. Accordingly, the classical Kendall-Smith coefficient of consistency measures deviations from transitivity in a tournament by counting the number of circular triads (directed $3$-cycles). In this paper, we characterize the fluctuations of the number of circular triads in inhomogeneous random tournaments and develop an inferential framework for the consistency coefficient. Specifically, we consider the $W$-random tournament model, where the comparison probabilities are determined by a tournamenton $W$, the analogue of a graphon in the tournament setting. We show that, for a $W$-random tournament on $n$ vertices, the number of circular triads exhibits three different fluctuation regimes, determined by suitable notions of regularity and uniformity of $W$. We further develop a novel tournamenton multiplier bootstrap that consistently approximates the limiting distribution of the circular-triad count in the relevant asymptotic regime. Combining this with procedures for testing regularity and uniformity, we design an algorithm for constructing confidence intervals for the consistency coefficient that is asymptotically valid for all tournamentons. We also obtain structural characterizations of tournamentons for which the limiting distribution of the number of circular triads exhibits specific degeneracies. These results can also be viewed through the lens of tournament quasirandomness and may be of independent interest.

For graphs $G$ and $F$, we say that $G$ is $F$-free if $F$ does not occur as an induced subgraph of $G$. This paper is concerned with the following question: Given an $F$-free graph $G$ having two vertices between which there exists at least one Hamiltonian path, how many Hamiltonian paths between these endpoints must exist (in terms of the order of $G$)? Our main result shows that there exists a sharp dichotomy. More precisely, we show that if $F$ is not an induced subgraph of $P_3+sP_1$ for any integer $s \geq 0$, then there exists an infinite family of $F$-free graphs having two vertices between which there exists a unique Hamiltonian path. On the other hand, we prove that if $F$ is an induced subgraph of $P_3+sP_1$ for some integer $s \geq 0$, then any $F$-free graph having two vertices between which there exists a Hamiltonian path contains exponentially many such paths between these two vertices. Our proofs use Ramsey-theoretic methods, a result on the existence of two vertices with low degree in graphs containing a unique Hamiltonian cycle, a path variant of Thomassen's red-independent weakly green-dominating sets, and a structural analysis of Hamiltonian paths in $P_3+sP_1$-free graphs. As an algorithmic consequence we obtain that for every fixed $s \geq 1$, given a Hamiltonian $sP_1$-free graph together with a Hamiltonian cycle, one can decide in linear time whether a second Hamiltonian cycle exists and construct one if it does.

Exact \(k\)-leaf powers are graphs whose edges are exactly the pairs of leaves at distance \(k\) in a tree. We prove explicit structure theorems for exact leaf powers on several representative graph families that test different exact-distance phenomena. Our most detailed root-classification theorem concerns chordless cycles: all exact \(5\)-leaf roots of \(C_\ell\), \(\ell\ge 8\), are described by a complete terminal block language. We also prove that the \(t\)-square ladder \(L_t\) is an exact \(5\)-leaf power if and only if \(t\le 2\). In contrast, dense bipartite square structures are often representable: among block-complete multipartite graphs, the exact \(5\)-leaf powers are precisely the bipartite members, and every bipartite co-cluster graph, including every crown \(K_{n,n}-M\), is an exact \(k\)-leaf power for every \(k\ge 5\). Finally, we give parity classifications for complete multipartite graphs and multipartite block graphs at larger exact distances, and isolate a sharp fan boundary at exact distance six.