组合数学

A coloring of the Hales--Jewett cube $[t]^n$ is symmetric if it is invariant under all coordinate permutations, and one-weight if it reads only an integer-weighted count of the letters. We prove that the two classes coincide -- a radix…

We give counterexamples to two well-known conjectures about matroids: White's conjecture on the generation of the toric ideal by symmetric exchange binomials, and a conjecture of Mason on the log-concavity of the counts of flats of a given…

We show that, for every fixed graph $H$, every $n$-vertex graph $G$ that excludes $H$ as a minor is $3$-colourable with clustering $O_H(n^{4/9})$. That is, there exists a function $f$ such that for every graph $H$, every $n\ge 1$, every…

Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$, and $\mathcal A (G)$ the set consisting of all minimal zero-sum subsequences over $G$. For any subset $\Omega \subset \mathcal F (G)$, we…

The geodesic treewidth of a graph $ G $ is the smallest $k$ for which there is a partition $\mathcal{P}$ into geodesics such that $G/\mathcal{P}$ has treewidth $k$, where $G/\mathcal{P}$ is obtained from $ G $ by contracting each part of $…

Inspired by Problem 17 from the 2024 American Mathematics Competition (AMC) 10B, this work focuses on enumerating the distinct outcomes of a snail race with specified number of ties of a certain type. We begin by developing a recurrence…

We provide a characterization of the connected subgraphs of the graphs with vertex set the non-isotropic points in a quadratic space $(V,Q)$, two points adjacent if and only if they span a tangent line. Here $(V,Q)$ is a quadratic space $V$…

The enumeration of planar maps with control on the boundary metric, i.e. the pseudometric induced on the outer face of the map by its bulk graph distance metric, is a difficult problem in general. However, we show that for a family of…

In 2020, Panda, Verma, and Keerti asked whether the central graph of every graph satisfies the AVD-total coloring conjecture. In this paper, we verify the conjecture for central graphs of regular graphs, complete bipartite graphs, graphs…

We study the one-time weight on strict partitions obtained from the modified odd Greaves--Jing--Zhu operator. The shifted $t$-Schur functions generated by this operator are obtained from the classical Schur $Q$-functions by the plethystic…

In this paper, we study the existence problem for spherical \(T\)-designs on the \(d\)-dimensional sphere, where \(T\) is an infinite subset of \(\mathbb N\). We show that, if \(d\ge 2\), then a finite subset of \(S^d\) has infinite…

Let $G$ be an $n$-vertex graph containing a Hamiltonian cycle and with minimum degree at least $3$. Gir\~{a}o, Kittipassorn and Narayanan (Israel J. Math., 2019) proved that $G$ contains another cycle of length at least $n-O(n^{4/5})$. In…

Mineyev's taiko construction, in Garg--Mineyev's finite support-size formulation, gives a concrete route from finite support data to zero divisors and units in group rings of torsion-free CAT(0) groups over $\mathbb{F}_2$. We prove that…

For graphs $F$ and $H$, let $\mathrm{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Very recently, Janzer, Longbrake, and Yepremyan proved that for $3<a\leq b$ and sufficiently large $t$,…

We construct doubly- and triply-graded Penrose-type homologies for ribbon graphs. The construction is a TQFT-valued cube of resolutions built from two-dimensional cobordisms, which may be nonorientable. Their Euler characteristics recover…

The $d$-Hoggatt triangle is a lower triangular matrix whose entries are given by specific minors of Pascal's triangle formed by consecutive $d$ rows and $d$ columns. The cases $d=1,2,3$ correspond to Pascal's triangle, the Narayana…

The transformation of the $h$-vector of a finite simplicial complex under an $\mathcal{F}$-uniform subdivision is encoded by a transformation matrix. Mu and Welker conjectured that the transformation matrix of the barycentric subdivision is…

For $m\in\mathbb{Z}_{\geq 0}$, let \[ N_{n,m}(x)={}_2F_1(-n,-n-m;m+1;x), \] which specializes to the Narayana polynomials of types $B$ and $A$ for $m=0$ and $m=1$, respectively. We prove that the associated basis transformation \[…

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family, $\Delta(\mathcal{F})=\max_{x\in[n]}|\{F\in\mathcal{F}:x\in F\}|$, and $\varrho(\mathcal{F})=\Delta(\mathcal{F})/|\mathcal{F}|$. Frankl and Wang conjectured that if $n>100k$…

We prove a sharp lower bound for the cardinality of sumsets of subsets of $\mathbb{Z}^d$ confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in…