Category: math.CO
Showing 1–20 of 77,609 results
Matt Larson, Ethan Partida
We prove the Hard Lefschetz theorem and Hodge-Riemann relations for certain rings which resemble the cohomology rings of projectivizations of globally generated vector bundles over toric varieties. This proves new cases of the standard conjecture of Hodge type and gives Bloch-Gieseker-type results for tautological classes of matroids.
Anton Molnar, Cosmin Pohoata, Michael Zheng
For every $n \geq 5$, we show that the Kneser graph of triangulations of a convex $n$-gon contains a Hamiltonian cycle.
Haixiang Zhang, Mengyu Cao, Mei Lu
For a family $\mathcal{H} \subseteq \binom{[n]}{k}$, a subset $\{A_1, A_2, \ldots, A_m\} \subseteq \mathcal{H}$ is called a \textit{matching} of size~$m$ if the sets $A_1, A_2, \ldots, A_m$ are pairwise disjoint. The \textit{matching number} of $\mathcal{H}$, denoted by $ν(\mathcal{H})$, is the largest integer~$m$ for which such a matching exists. $\{A_1,A_2,\ldots,A_l\}\subseteq \binom{[n]}{k}$ is said to be a \textit{$k$-uniform sunflower} with $l$ \textit{petals}, if there exists a core set $C\subseteq[n]$ contained in every $A_i$ and $A_i\setminus C$ are pairwise disjoint, for $1\leq i\leq l$. Let $S_{k,l}^{k-1}$ denote the $k$-uniform sunflower with $l$ petals and the core set of size $k-1$. The \textit{codegree} of $E$ in $\mathcal{H}$, denoted by $d_{\mathcal{H}}(E)$, is defined as $d_{\mathcal{H}}(E) =|\{F\in \mathcal{H}:E\subseteq F\}|$. Let the \textit{$\ell_p$-norm} of $\mathcal{H}$ be $co_p(\mathcal{H})= \sum_{E\in \binom{[n]}{k-1}}(d_{\mathcal{H}}(E))^p$. For sufficiently large $n$, we determine the maximum $\ell_p$-norm and the maximum number of sunflowers $S_{k,l}^{k-1}$ for a family $\mathcal{F} \subseteq \binom{[n]}{k}$ with matching number $ν(\mathcal{F}) = s$. These results can be viewed as a Turán-type problem (specifically $\mathrm{ex}_k(n, S_{k,l}^{k-1}, M_s)$) and a generalization of the Erdős Matching Conjecture. Furthermore, for the case $k = 3$, we establish a linear threshold for $n$.
Mudit Aggarwal, Hrishik Koley, Samrith Ram
We derive explicit rational generating functions for weighted tilings of $2k\times n$ rectangles by straight $k\times 1$ tiles. Our approach combines a decomposition by fault lines with a Hadamard-product framework. Tools from algebraic combinatorics are used together with a theorem of Klivans and Reiner on Schur expansions of plethystic compositions of elementary symmetric functions. This translates the tiling problem into a combinatorial framework via special rim-hook tableaux. On the tiling side, Graham's theorem on fault-free tilings provides the key input needed to complete the analysis.
Cheng Chi, Long-tu Yuan
We study edge-colorings of the complete $p$-graph on $n$ vertices that contain no three edges $A,B,C$ of distinct colors such that the symmetric difference of $A$ and $B$ is contained in $C$. For $p\ge3$ and $n\ge p+1$, we show that every such coloring contains at most $1+\floor{n/p}$ colors and characterize the extremal colorings, generalizing a theorem of Erdős, Simonovits and Sós. %\cite{erdos1975}. When $p=3$, the condition $A\triangle B\subseteq C$ implies $|A\triangle B|=2$, and the three edges necessarily form a copy of $F_4\coloneqq\{abc,abd,bcd\}$ or $F_5\coloneqq\{abc,abd,cde\}$. For $n\ge5$, we show that every rainbow $F_5$-free edge-coloring is rainbow cancellative. For rainbow $F_4$-free colorings, we construct colorings with $m(n)+1$ colors for all $n\ge4$, where $m(n)$ is the size of a maximum partial Steiner triple system of order $n$ and satisfies $m(n)=n^2/6+O(n)$, improving the linear lower bound by Budden and Stiles. %\cite{budden}. Moreover, for $n=2^s-1$, we obtain $\ar(n,F_4)\ge m(n)+n^2/42+o(n^2)=4n^2/21+o(n^2)$ via a construction based on independent sets in the Grassmann graph. We also prove that $\ar(n,F_4)\le (5n^2-8n)/21$ for $n\ge4$, improving the quadratic coefficient in the upper bound of Budden and Stiles from $1/4$ to $5/21$.
Lies Beers, Raffaella Mulas
We investigate Fair and Tolerant (FAT) graph colorings, a coloring framework in which each vertex is allowed to share its color with a prescribed fraction of its neighbors, while the remaining neighbors are required to be distributed evenly among the other coloring classes. In particular, we determine the FAT chromatic number for all complete multipartite graphs, and we analyze the behavior of FAT colorings under several graph operations. Although spectral methods form the primary focus, several combinatorial arguments are included to complement the results.
Simona Bonvicini, Tomaž Pisanski, Arjana Žitnik
A bicirculant is a regular graph that admits an automorphism having two vertex-orbits of the same size. A bicirculant can be described as follows. Given an integer $m \ge 1$ and sets $R, S, T \subseteq \mathbb Z_m$ such that $R=-R$, $T=-T$, $0 \not\in R \cup T$ and $0 \in S$, the graph $B(m;R,S,T)$ has vertex set $V=\{u_0,\dots,u_{m-1},v_0,\dots,v_m-1\}$ and edge set $E=\{u_iu_{i+j}| \ i \in\mathbb Z_m, j \in R\} \cup \{v_iv_{i+j}| \ i \in\mathbb Z_m, j \in T\} \cup\{u_iv_{i+j}| \ i \in\mathbb Z_m, j \in S\}.$ Bicirculant graphs with $R=T=\emptyset$ are known as cyclic Haar graphs. In 2025 we conjectured that the only non-hamiltonian graphs among regular connected bicirculants of degree more than one are the generalized Petersen graphs $G(m,2)$ with $m \equiv 5 \pmod 6$. Recently we have verified the conjecture for bicirculants with $|S|\le 2$ and for bicirculants with $|R|=|T|$ odd. In this paper we show that the conjecture holds for all bicirculants with $|S| \le 3$ and for all bicirculants with $|S| \ge 4$ and $m/\gcd(m, S)$ even. As a byproduct of our results, we prove that every connected bicirculant graph on $2m$ vertices with $|S| \ge 4$ is hamiltonian for even $m< 9\, 240$, and for odd $m< 3\,465$. Finally, we show that the existence of a hamilton cycle in every connected cyclic Haar graph of valence at least $4$ implies that every connected bicirculant graph of valence at least $4$ is hamiltonian.
Licheng Zhang, Yuanqiu Huang, Fengming Dong
In 2016, Dowden initiated the study of planar Turán-type problems, which has since attracted considerable attention. Recently, Bekos et al. proved that every $K_3$-free $1$-planar graph on $n\ge 4$ vertices has at most $3n-6$ edges. In this paper, we strengthen this bound to $3n - 8$, which is tight for all even $n \ge 8$. Furthermore, we show that every $K_4$-free $1$-planar graph on $n \ge 3$ vertices has at most $\bigl\lfloor \tfrac{7n}{2} \bigr\rfloor - 7$ edges, and this bound is tight for all integers $n \ge 9$. We also prove that every $K_5$-free $1$-planar graph on $n \ge 3$ vertices has at most $4n - 8$ edges, which is tight for $n = 8$ and for all integers $n \ge 10$.
Andreas Holmsen, Attila Jung, Balázs Keszegh, Dániel G. Simon, Gábor Tardos
Graphs whose maximum clique size exceeds half of the total number of vertices satisfy a classical property: the family of their maximum sized cliques can be pierced by a single vertex. This result dates back to a 1965 theorem by Hajnal. Motivated by this theorem, Jung, Keszegh, Pálvölgyi, and Yuditsky recently conjectured that an analogous result should hold for hypergraphs of larger uniformity, with an appropriate constant replacing the threshold $1/2$. In this paper we refute this conjecture in a strong form. We show that for any constant $c<1$ and integers $k\ge 3$ and $t\ge 1$, there exist $k$-uniform hypergraphs $G$ whose maximum clique size exceeds $c|V(G)|$, yet the family of maximum size cliques of $G$ cannot be pierced by $t$ vertices. This demonstrates that no universal constant threshold guarantees bounded piercing number for maximum cliques in uniform hypergraphs. We discuss further questions concerning the relationship between clique size and piercing maximum cliques in hypergraphs, and introduce a geometric variant of the problem using Helly's Theorem.
Ronen Wdowinski
For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a reconfiguration graph on independent transversals of $(G,\mathcal{U})$ is connected, meaning any independent transversal can be transformed into any other one through a sequence of one-vertex modifications while always maintaining an independent transversal. Analogous to a theorem of Haxell, they proved that this is the case if $G$ has maximum degree $Δ$ and each block of $\mathcal{U}$ has size at least $2Δ$, except if the union of some $k \ge 1$ blocks of $\mathcal{U}$ induces $k$ disjoint copies of the complete bipartite graph $K_{Δ, Δ}$ in $G$. Solving one of their problems, we exactly characterize the partition structure in the latter exceptional instances of their theorem, showing that there is a rich variety of them but they are generated by a simple constructive procedure.
Sara Koljančić, Polona Oblak
In this paper, we investigate the non-symmetric Strong Spectral Property (nSSP) from a combinatorial perspective. To zero-nonzero patterns of matrices we associate directed graphs and study when they require or allow the nSSP, providing a framework that avoids verifying the nSSP for individual matrices. A new combinatorial method is introduced and used to recognise several patterns that require the nSSP. It is shown that loop assignments in double paths play a critical role in establishing this property, and we show that an open question regarding irreducible tridiagonal patterns has a negative answer. We also investigate whether the minimum number of arcs in a directed graph on $n$ vertices that requires the nSSP, is equal to $2n-1$, and confirm this minimum for several specific digraph families.
Yichen Wang, Mengyu Duan, Dániel Gerbner, Hilal Hama Karim
Given a hypergraph $F$, what is the largest chromatic number that an $F$-free hypergraph can have? In the case of graphs, this question is easy to answer: the chromatic number is unbounded if $F$ contains a cycle, and the largest chromatic number of $F$-free graphs is $k-1$ if $F$ is a forest on $k$ vertices. The situation is more complicated for hypergraphs. The strong coloring of a hypergraph is a coloring of the vertices such that every hyperedge is rainbow. The weak coloring of a hypergraph is a coloring of the vertices such that no hyperedge is monochromatic. The strong/weak chromatic number of a hypergraph is the minimum number of colors in a strong/weak coloring of the hypergraph. Our question has been completely answered for the weak chromatic number, similarly to the graph case. We characterize the hypergraphs $F$ such that $F$-free hypergraphs have bounded strong chromatic number. The only remaining case is when $F$ is the 3-uniform expansion $S_k^+$ of a star with $k$ edges. Concerning the strong chromatic number of $S_k^+$-free hypergraphs, we give bounds that are asymptitically sharp as $k\rightarrow\infty$. We also consider the same problem when the Berge copies of a graph $F$ are forbidden. We characterize when the strong/weak chromatic numbers are bounded in this case, and obtain sharp results or bounds for specific trees. In particular, when $F$ is a path, we give a tight bound when $r=3$ and an asymptotically sharp bound when $r=4$.
Jorik Jooken, Denys Lohvynov
A $k$-regular graph of girth $g$ is called vertex-girth-regular if every vertex is contained in the same number of cycles of length $g$. For integers $n, k, g$ and $λ$, we denote such a graph on $n$ vertices in which every vertex lies on exactly $λ$ cycles of length $g$ by a $\text{vgr}(n,k,g,λ)$-graph. It is well-known that any vertex-girth-regular graph satisfies $λ\le \frac{k(k-1)^{\left\lfloor \frac{g}{2} \right\rfloor}}{2}$. Graphs for which $λ$ is close to this bound are of particular interest in connection with the cage problem, since requiring many girth cycles through every vertex is a natural way to isolate highly structured candidates for small regular graphs of prescribed girth. In this paper, we prove that for every $k\ge 3$ and every integer $0< \varepsilon \leq \frac{k-1}{2}$, there does not exist a $\text{vgr}(n,k,5,\frac{k(k-1)^2}{2}-\varepsilon)$-graph. Previous non-existence results had already settled all odd girths at least $7$ and very recently also girth $3$, leaving girth $5$ as the only girth for which no non-trivial non-existence result was known. Thus, our result resolves the final remaining case and completes the picture for odd girths.
Yongbin Gao, Ligong Wang
Let $κ'(G)$ and $τ(G)$ denote the edge connectivity and the spanning tree packing number of a graph $G$, respectively. Recently, Zhang and Fan [J. Graph Theory 112 (2) (2026) 128--144] posed the problem of finding a tight spectral radius condition for an $m$-edge-connected graph $G$ to guarantee $τ(G)\ge k$ for $k+1\le m\le 2k-1$, and solved the case $k=2$ and $m=3$. In this paper, we study the general case $m=k+1$ for $k\ge 3$. We obtain a tight spectral radius condition for a $(k+1)$-edge-connected graph to contain $k$ edge-disjoint spanning trees, and characterize the corresponding extremal graph. More precisely, we prove that if $k\ge 3$ and $G$ is a $(k+1)$-edge-connected graph of order $n\ge 3k+2$, and $ρ(G)\ge ρ(F_{n,k})$, then $τ(G)\ge k$ unless $G\cong F_{n,k}$, where $F_{n,k}$ is the graph with vertex set $U\cup T$, in which $U=\{u_1,u_2,\ldots,u_{n-4}\}$ and $T=\{v_1,v_2,v_3,v_4\}$, such that $F_{n,k}[U]\cong K_{n-4}$, $F_{n,k}[T]\cong K_4-e$ with missing edge $v_1v_2$, and $E_G(T,U)=\{v_i u_j : i \in \{1,2\},\ 1 \le j \le k-1\} \cup \{v_i u_j : i \in \{3,4\},\ 1 \le j \le k-2\}$. We also give a heuristic discussion of the more general case $m=k+c$ with $0\le c\le k-1$ and propose a conjecture on the structure of the corresponding extremal graphs.
Alex Iosevich, Vishal Gupta
It is well-known in industrial data science that large values of real-life time series tend to be structured and often follow concrete and visible patterns. In this paper, we use ideas from additive combinatorics and discrete Fourier analysis to give this heuristic a mathematical foundation. Our main tool is the Fourier ratio, a complexity measure previously used in compressed sensing, combined with a generalized version of Chang's lemma from additive combinatorics. Together, these yield a precise prediction: when the Fourier ratio of a time series is small, the set of its largest values can be additively generated by a very small set using only $\{-1,0,1\}$ coefficients. We test this prediction on US inflation data and Delhi climate data, both in their original form and after mean-centering. The numerical results confirm the predicted structure: a generating set of size $4$--$7$ suffices to span large spectra containing dozens of points, even when the Fourier ratio is large enough that our theoretical bounds become loose. These findings provide a rigorous explanation for why extreme values in real-world data are information-rich and structurally significant.
Benjamin Przybocki, John Mackey, Marijn J. H. Heule, Bernardo Subercaseaux
Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to generate and formalize correctness proofs in Lean. This case study highlights the potential of integrating automated reasoning, large language models, and formal verification to accelerate mathematical discovery. We argue that such tool-driven workflows will play an increasingly central role in experimental mathematics.
Huawei Wu, Lewen Wang, Sihuang Hu
We present a uniform framework for constructing \(3\)-designs from \(\mathrm{GL}_2(\mathbb F_q)\)-invariant subspaces of \(\mathbb F_q[X,Y]_k\), the space of homogeneous polynomials of degree \(k\). Given such a subspace \(W\), we associate a \(\mathrm{PGL}_2(\mathbb F_q)\)-invariant family of \(k\)-subsets of \(\mathbb P^1(\mathbb F_q)\). Whenever this family is nonempty, it forms a \(3\text{-}(q+1,k,λ)\) design. When \(k\le q\), the evaluation map on \(\mathbb P^1(\mathbb F_q)\) identifies \(W\) with a subcode \(C_W\) of the projective Reed--Solomon code. We also show that the supports of minimum-weight codewords in \(C_W\), as well as the supports of suitable fixed-weight codewords in the dual code \(C_W^\perp\), yield further \(3\)-designs. Via the Cayley transform, the construction is transferred to the unit circle \(U_{q+1}\subseteq \mathbb F_{q^2}^{\times}\), where the block conditions become explicit linear relations among elementary symmetric polynomials. Applying this framework to the Lucas subspaces, we obtain explicit block descriptions, classify the cases in which the defining conditions reduce to a single equation, and establish several emptiness and nonemptiness results. In particular, for \(q=p^e\) and \(k=p^m+1\), we show that the associated block family is nonempty if and only if \(m\mid e\), in which case it yields the Steiner system \(S(3,p^m+1,q+1)\). Finally, in the ternary case \(p=3\) and \(k=7\), we use the weight distribution of the ternary Melas code to determine the design parameters left undetermined by Xu et al. (Designs, Codes and Cryptography: Vol. 92, 2024).
Jade S. Davies, Peter J. Dukes
Let $n=hw$, where $h$ and $w$ are integers with $h,w \ge 2$. We determine the set of possible intersection numbers of two $n \times n$ latin squares having the additional `Sudoku' constraint based on a $w \times h$ grid of $h \times w$ boxes.
Brian DSouza, Jessica Pereira
We study additively graceful labelings of signed graphs on stars and double stars. While the case of signed stars is straightforward, the problem becomes significantly more intricate for signed double stars. We obtain a characterization of additively graceful signed stars, while, for several sub-classes of additively graceful signed double stars, we establish existence, uniqueness and non-existence results.
Sébastien Labbé, Peter Selinger
We present a simple construction of hat tilings. The construction can be carried out by superimposing a triangular grid on a specially colored image and reading off the orientation of the tiles. We show that our construction produces valid hat tilings, and conversely, in an appropriate sense that is made precise in the paper, that every valid hat tiling can be obtained in this way.