组合数学

A partial Latin square of order $n$ is called $\epsilon$-dense if each row and each column contains at most $\epsilon n$ filled cells, and each symbol occurs at most $\epsilon n$ times. A partial Latin square is said to be completable if…

Let $r\ge 3$ and let $1>p_1\ge p_2\ge\cdots\ge p_n>0$. Let $\mu_{\mathbf p}$ denote the product measure on $2^{[n]}$ where each coordinate $i$ is included independently with probability $p_i$. A family $\mathcal A\subseteq 2^{[n]}$ is…

A factor of a graph is essentially a specific type spanning subgraph. In recent years, the spectral extremal problem of characterizing the existence of graph factors via eigenvalues has been widely studied. This paper focuses on fractional…

In this paper, we investigate the effective resistance on the graph $G_N^{(r)}$, which is obtained by deleting all edges corresponding to circular distances $\{\pm1, \pm2, \dots, \pm r\}$ from the complete graph $K_N$. We utilize the cyclic…

We derive several applications of the path-minimality theorem for adjacency $p$-energy proved in the companion paper. First, we prove the sharp inequality $$ \mathcal E_p^+(G)\ge \mathcal E_p^+(P_n), $$ where $P_n$ is the path on $n$…

Let $\Omega$ be the superspace ring of regular differential forms on the affine space $\mathbb{C}^n$. If $G \subseteq GL_n(\mathbb{C})$ is a complex reflection group, the {\em $G$-superspace coinvariant ring} is the quotient $SR_G :=…

Throttling is a graph optimization problem, where the throttling number of a graph is the minimum sum or minimum product of the number of vertices in an initial set and the time required to complete a certain graph operation. A…

We derive sparse analogs of several Roth-type results, showing that they hold in $B_h$ sets of near-maximum size. It is shown that if a $B_h$ set is free of pairwise distinct solutions to a linear equation with more than $2h$ variables then…

We show that, when $d=o(n)$, every $d$-regular $n$-vertex graph contains a spanning subgraph whose degree distribution is nearly uniform, i.e., for each $0\leq i\leq d$, there are $(1+o(1))n/(d+1)$ vertices with degree $i$. This proves a…

Naslund and Sawin used the slice-rank method for diagonal tensors to prove that $$|\mathcal{F}|=O\!\left(n^{1/2}\left(\frac{3}{2^{2/3}}\right)^n\right)$$ for any sunflower-free family $\mathcal{F}\subseteq 2^{[n]}$. We prove a lemma similar…

We prove Seymour's second neighborhood conjecture on oriented graphs whose minimum out-degree is equal to $7$. This gives, to our knowledge, the first improvement of the minimum out-degree threshold in two decades, since the work of Kaneko…

For $r \geq 2$ and graphs $H_1, \ldots, H_r, G$, we say that $G$ is $(H_1, \ldots, H_r)$ vertex-Ramsey, or $(H_1, \ldots, H_r)_v$-Ramsey, if whenever we colour the vertices of $G$ with colours from the set $[r]=\{1,2, \ldots, r\}$ there…

Let $G$ be a finite group with $|G|=p^m$ where $p$ is a prime and $m$ is a positive integer. Let $k<p$. Let $a_1,\ldots,a_k\in G$ be pairwise distinct and let $b_1,\ldots,b_k\in G$. Then there exists a permutation $\sigma$ on $1,\ldots,k$…

Burr, Erd\H os, Graham, and S\'os introduced the maximal anti-Ramsey function $\chi_{\mathrm{S}}(n,e,L)$, the minimum number of colors required over all $n$-vertex graphs with at least $e$ edges such that every copy of $L$ is rainbow. In…

Let $P=\{x_1,\ldots,x_n\}$ be a multiset consisting of $n\ge 2$ real numbers such that $\sum_{i=1}^{n}x_i=0$ and $\sum_{i=1}^{n}|x_i|>0$, and let $k <n$ be a positive integer. We sample $k$ elements from $P$ without replacement and set…

Given a set $R$ of positive integers, an $R$-graph $H = (V, E)$ is a hypergraph where the cardinality of each hyperedge belongs to $R$. If $R = \{r\}$, we sometimes refer to the hypergraph as an $r$-graph rather than an $R$-graph. For a set…

In this paper we prove a generalisation of the concentration-of-measure phenomenon in the discrete cube. In this setting, the concentration-of-measure phenomenon states that for every subset $\mathcal{A}$ of the discrete cube, its sum with…

We show that list $3$-coloring a~$C_4$-free graph of diameter-$2$ can be done in polynomial-time. Our algorithm is based on a structural characterization showing that many such graphs are not~$3$-colorable. In particular, we show…

A graph $G$ is $(t+1)K_{r+1}$-free if it contains no $t+1$ pairwise vertex-disjoint copies of $K_{r+1}$. Moon [Canad. J. Math. 20 (1968) 95-102] and Simonovits [Theory of Graphs (Proc. Colloq., Tihany, 1966)] independently determined that,…

We study the two-color distinct-part series \(S_1(q)\), equivalently Andrews' generating function \(v_d(q)\) for strictly concave compositions, and its odd and even companions \(T_o(q)\) and \(T_e(q)\). We determine the coefficients of…