组合数学

We introduce a binary matroid approach to the enumeration of mod 2 toric-colorable seeds of fixed Picard number. We organize these matroids by their contraction category and enumerate weak pseudomanifold subcomplexes by a dynamic…

Let $A$ be a finite non-empty subset of an abelian group $G$, and let $r_A(d)=|\{(a,a')\in A^2:a-a'=d\}|$. Croot and Lev asked whether the pointwise half-threshold condition $r_A(d)\ge |A|/2$ for every $d\in A-A$ forces $A-A$ to be either a…

In this note, we give a Tur\'an theorem for Cayley graphs $\Cay(\Z_p,S)$ over prime cyclic groups $\Z_p$. For a graph $F$ and a finite abelian group $G$, define the Cayley--Tur\'an number by \[ \exCay(F,G) = \max\{|S|:S=-S\subseteq…

An $(n,R)$-covering sequence over a finite alphabet $\Sigma_q = \{0,1,\dots, q-1\}$ is a cyclic sequence whose consecutive length-$n$ windows form a covering code of radius $R$. Equivalently, every word in $\Sigma_q^n$ is within Hamming…

For an integer $q\ge 2$ and a graph $F$ satisfying $q\mid e(F)$, the zero-sum Ramsey number $R(F,\mathbb Z_q)$ is the least integer $n$ such that every edge-labeling $w\colon E(K_n)\to \mathbb Z_q$ contains a copy of $F$ whose edge-label…

Given a graph $G$, the domination number $\gamma(G)$ is the minimum cardinality of a dominating set in $G$, and the packing number $\rho(G)$ is the maximum cardinality of a set of vertices that are pairwise at distance at least $3$. The…

The Moore graph of degree $57$, if one exists, is the remaining open case of the Hoffman-Singleton classification in diameter two. Although its existence remains open, substantial restrictions on the automorphism group of such a graph are…

Andrews and El Bachraoui investigated two-color partitions in which even parts may occur only in blue and obtained several identities relating parity refinements of these partitions to overpartitions and minimal excludants. Some of their…

We discuss the problem of characterizing equidistant binary codes of a given length $n$ having largest possible distance and the maximum number of codewords. Such characterizations have been studied by several authors over the years and…

We model Herman Kahn's escalation ladder as an impartial combinatorial game. Reindexing each rung by its distance to the nuclear threshold turns the ladder into a subtraction game, the most tractable class in combinatorial game theory, and…

A subgraph $H$ of $G$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph…

In this paper, we study the so-called log-convergence of graphs defined by Bal\'azs Szegedy (arXiv:1504.00858). We answer his Question 4 affirmatively: the sequence of incidence graphs of projective planes over finite fields log-converges,…

The Bilu-Linial conjecture asserts that every $d$-regular graph admits a signing $\sigma$ such that the spectral radius of the signed adjacency matrix $A_\sigma$ satisfies $\rho(A_\sigma)\le 2\sqrt{d-1}$. Bilu and Linial also proved the…

A graph $G$ is said to be $F$-semi-saturated if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. The semi-saturation number $ssat(n,F)$ is the minimum number of edges in an $F$-semi-saturated graph of…

For a strict partition $\lambda$, let $\mathcal Q_\lambda(X;t)=Q_\lambda[X-tX]$ be the shifted $t$-Schur function arising from the modified Greaves--Jing--Zhu operator on the odd power-sum ring. We study transition matrices between the…

Let $T(x)=\prod_{k=0}^{\infty}(1-x^{2^k})$ be the generating function of the Thue--Morse sequence, and write $T(x)^m=\sum_{n\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \[…

We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form $\mathrm{conv}(W \cdot \mathbf{a})$ where $W$ is a finite Coxeter group acting on $\mathbb{R}^n$ and…

Let $\gamma \ge 1$. A set $A$ of nonnegative integers is a Sidon set if for each $d>0$ there is at most one pair $(a,b) \in A \times A$ with $d=a-b$. If there are at most $\gamma$ pairs, then $A$ is a $\gamma$-Golomb ruler. We prove that if…

More than forty years ago, Andersen and Hoffman independently proved that every symmetric Latin rectangle can be extended to a symmetric Latin square with prescribed diagonal entries. We generalize this theorem as follows. Let $k\leq n^2$,…

Cayley codes, introduced by Kaufman and Wigderson, are linear codes constructed from a Cayley graph and a smaller linear code. We explore general properties of the class of Cayley codes for finite groups. In particular we give a reduction…