组合数学

We introduce a colorful version of separating path systems, in which two edges can only be separated from each other by two paths of distinct colors. We calculate the minimum sizes of such systems for various standard classes of graphs and…

In 1981, Erd\H{o}s and Faudree asked whether there exists an infinite family of graphs $G_N$ on $N$ vertices with $\Delta(G_N)<N-1$ and $\sri(G_N)=1$, and whether every family with $|V(G_N)|=N$ and $\Delta(G_N)<c$ for some fixed constant…

The general position number for graphs ask for largest vertex subsets $S$ such that no three vertices are contained on a common shortest path. We examine this problem in the setting of directed graphs. We provide bounds for the general…

An {\em odd subgraph} of a graph is a subgraph in which every vertex has odd degree. A graph $G$ is said to be {\em odd $k$-edge-colorable} if there exists an edge-coloring $E(G) \rightarrow \{1,2, \ldots, k\}$ such that each non-empty…

The Tur\'an number of a graph $F$, $ex(n,F)$, is the maximum number of edges in a graph on $n$ vertices which does not contain $F$ as a subgraph. Let $S_{a,b}$ denote a double star with a central edge $uv$, $a$ leaves connected to $u$ and…

For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $\lambda_1(G) \geq \lambda_2(G) \geq \dots \geq \lambda_n(G)$ be its eigenvalues. For an integer $p \geq 2$, the positive $p$-energy and…

It is proved that for a graph of order $n$, where $n\ge 6$, if the signless Laplacian index is larger than or equal to certain value depending on $n$, then the graph contains a trebly chorded cycle, where the chords incident to a common…

Researchers have repeatedly found that the ends of an RNA sequence are significantly closer than expected for a random linear chain. However, we prove that the ends of a branched structure are almost certainly close. Our results are…

We study clustering through the partitions it induces on a finite labeled set $[n]=\{1,\dots,n\}$, and analyze how these partitions change under perturbations of a point configuration $X=(x_1,\dots,x_n)\in(\mathbb{R}^d)^n$. We equip the…

In this paper, we are motivated by the conjectures proposed by C.~Bender \textit{et al.}, \cite{C} in 2024. We have settled the first two conjectures negatively by providing a counter example in \cite{KTJ}, whereas in this paper, we prove…

The $r$-neighbour bootstrap process on a graph $G$ begins with a set of infected vertices; subsequently, healthy vertices become infected once they have at least $r$ infected neighbours. The central extremal problem in bootstrap percolation…

In this paper, we derive a set of equivalent conditions for the zero-divisor graph $\Gamma(Q)$ of a poset $Q$ with $0$ to be complemented, characterizing it in terms of quasi-complemented posets. Furthermore, we prove that the notions of a…

In this note, we prove that any 2-coloring of the plane contains 4 points of the same color forming a rhombus with unit sides and non-unit diagonals, answering a question of Axenovich, Liu, and the second author.

The wheel $W_{k}$ is the graph on $k+1$ vertices consisting of a vertex joined to a cycle of length $k$, and we say that $W_k$ is an even wheel if $k$ is even. Mao, Wang, Magnant, Schiermeyer proved that the Ramsey number of $W_{2n}$ is…

Relying on the Classification of Finite Simple Groups it was shown by Feng and Xu (Discrete Math., 2005) that every quartic Cayley graph of a regular $p$-group, $p \neq 2,5$, is normal. In this paper a CFSG-free proof of Feng-Xu theorem is…

Given a pair of finite posets $A \subseteq P$, the function counting integer-valued order preserving extensions of an order preserving map $\lambda : A\rightarrow \mathbb{Z}$ from $A$ to $P$ is given by a piecewise polynomial in $\lambda$.…

Let $G$ be a finite simple graph with oriented incidence matrix $D$. The signed graph on edge set $E(G)$ with adjacency matrix \[ A_{\mathcal A(G)}=D^{\mathsf T}D-2I \] is classical in the signed-line-graph literature. In this paper we…

In this paper, we prove that the zero-divisor graph $\Gamma(P)$ of a Boolean poset $P$ is both well-covered and Cohen--Macaulay. Furthermore, for a poset $\mathbf{P} = \prod_{i=1}^{n} P_i$ $(n \ge 3)$, where each $P_i$ is a finite bounded…

In this work, we investigate Maker-Breaker directed triangle games, a directionally constrained variant of the classical Maker-Breaker triangle game. Our board of interest is a tournament, and the winning sets are all $3$-cycles present in…

We study the doubled edge-stage lift \[ \HL'_2(G)=L(G\otimes K_2), \] the line graph of the canonical bipartite double cover of a graph \(G\). The natural involution \((u,v)\leftrightarrow(v,u)\) has quotient isomorphic to \(L(G)\), and…