组合数学
For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $\chi(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2\chi(G))$…
Juniper Green is a simple combinatorial game invented by Rob Porteous and popularized by Ian Stewart. It was originally designed to familiarize school children with the concepts of multiplication and division. We analyze this elementary…
Cyclic equalizability is a notion introduced by Shinagawa and Nuida in 2025, in the study of card-based cryptography. Informally, a collection of words is cyclically equalizable if, by inserting the same letters at the same positions in all…
In this paper, we determine the graphs with maximum value of the sum number from $k$-clique spectral radius to $(2r-1)$-clique spectral radius among all $2K_{r}$-free graphs on $n$ vertices for $ r\le k$ and large $n$. We also determine the…
Corsten and Frankl conjectured that a simplex is diameter-Ramsey if and only if its circumcenter lies in its convex hull. We disprove this conjecture in every dimension $d\ge 3$. The main tool is a sufficient criterion based on a…
Linek's 1989 problem asks whether the numbers of independent sets of trees avoid infinitely many positive integers. We show that the set of natural numbers realized as the number of independent sets of a tree has a lower growth exponent of…
For a graph $G$, let $\mathscr{C}_5(G)$ denote the graph whose vertices are the induced $5$-cycles of $G$, where two vertices are adjacent whenever the corresponding cycles share an edge. We investigate the iterative behavior of the…
Our first main result shows that, for words with a fixed multiset of weak right-to-left minima, the statistics within each of the following three classes are equidistributed: 1. Mahonian statistics: $\textsf{inv}$, $\textsf{maj}$,…
This paper investigates a special variant of a pursuit-evasion game called lions and contamination. In a graph where all vertices are initially contaminated, a set of lions traverses the graph, clearing the contamination from every vertex…
Given $n \geq 1$, we study the existence of a tree on $n$ vertices whose independence polynomial is symmetric and unimodal as well as the existence of a symmetric and unimodal independence polynomial of degree $n$ of a tree.
We investigate the inner vertex-isoperimetric problem on the $d$-regular tree $T_d$. We first determine the exact value of the inner vertex-isoperimetric profile $I_d(k) = \min\{ |\partial D| \mid D\subset T_d \text{ finite and connected},\…
The graph $G_\sigma$ is obtained from graph $G$ by attaching self loops on $\sigma$ vertices. The energy $ E(G_\sigma)$ of the graph $G_\sigma$ with order $n$ and eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$ is defined as $…
Defant and Zheng introduced a consecutive-pattern-avoiding stack sort map $SC_{\sigma}$, where the stack must avoid a consecutive pattern $\sigma$. Seidel and Sun disproved a conjecture in Defant and Zheng's paper about the maximum…
A chain is defined as a directed acyclic graph (DAG) with one source and one sink, where the children are ordered and the spanning tree computed using a depth-first search is a path. Such DAGs emerge in the context of tree compression and…
It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On…
We present improved lower bounds for nine classical Ramsey numbers: $\mathbf{R}(3, 13)$ is increased from $60$ to $61$, $\mathbf{R}(3, 18)$ from $99$ to $100$, $\mathbf{R}(4, 13)$ from $138$ to $139$, $\mathbf{R}(4, 14)$ from $147$ to…
Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family. For an element $i\in[n]$, the degree of $i$ is the number of sets in $\mathcal{F}$ that contain $i$. Assume that the degrees are ordered as $d_{1}\ge d_{2}\ge\cdots\ge…
For a group $G$ and subsets $S,T \subset G$ we introduce the mirror di-Cayley graph $MX(G;S,T)$ and mirror di-Cayley sum graph $MX^+(G;S,T)$ with connections sets $S$ and $T$ (MDCGs for short). We refer to them indistinctly by…
We present a new formula for umbral operators that yields three main insights. First, it makes explicit a connection between umbral calculus and iteration theory. Second, it leads naturally to a definition of fractional exponents of umbral…
We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a…