On graph coloring check-digit method

完全不相交的三位十进制校验码

Totally Disjoint 3-Digit Decimal Check Digit Codes

本研究提出3个两两不相交的三位十进制检错码，这类码具备无排列特性，可消除循环错误与语音错误，推进了Verhoeff提出的十进制校验码相关研究。

一种无排列的长度为3的十进制校验位码

A Permutation-Free Length 3 Decimal Check Digit Code

本文构造了一种长度为3的十进制校验位码，任意两位数字即可确定整个码字，从而消除循环错误和语音错误。

Every 4-regular graph is acyclically edge-6-colorable

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. Fiam${\rm \check{c}}$ik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that $a'(G)\le Δ+ 2$ for any simple graph $G$ with maximum degree $Δ$. Basavaraju and Chandran (2009) showed that every graph $G$ with $Δ=4$, which is not 4-regular, satisfies the conjecture. In this paper, we settle the 4-regular case, i.e., we show that every 4-regular g…

Injective Edge Chromatic Index of a Graph

Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that if $e_{1}, e_{2}$ and $e_{3}$ are consecutive edges in $G$, then $c(e_{1})\neq c(e_3)$. The injective edge coloring number $χ_{i}^{'}(G)$ is the minimum number of colors permitted in such a coloring. In this paper, exact values of $χ_{i}^{'}(G)$ for several classes of graphs are obtained, upper and lower bounds for $χ_{i}^{'}(G)$ are introduced and it is proven that…

New Upper Bounds for the Erdős-Gyárfás Problem on Generalized Ramsey Numbers

A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ which assigns at least $q$ colors to each $p$-clique. The problem of determining the minimum number of colors, $f(n,p,q)$, needed to give a $(p,q)$-coloring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$. The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in\{4,5\}$, each of which was proved by giv…

无环着色参数的改进界

Improved bounds for acyclic coloring parameters

本文研究了图的无环着色参数，证明了无环色指数最多为$2Δ-1$，并在最大度$Δ$足够大时给出无环色数的新上界，通过改进的随机算法在非正常着色基础上逐步实现正常着色以节省颜色。

Khovanov同调、Rasmussen的s-不变量与4-球中奇异圆盘的生成树模型

A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball

本文为Khovanov同调的生成树复形明确定义了微分形式，描述了Rasmussen的s-不变量，并据此发现了一个在4-球中界定奇异圆盘的无限纽结族。

关于次立方外平面图的碎屑着色猜想

The crumby coloring conjecture for subcubic outerplanar graphs

研究次立方外平面图的碎屑着色猜想，引入有根文法，将碎屑着色扩展为可接受碎屑的着色，通过计算机辅助验证和手工证明，得出每个有限简单次立方外平面图都允许碎屑着色的结论。

灵活实现的存在性：关于稀疏图的NP完全性及算法

Flexible realizations existence: NP-completeness on sparse graphs and algorithms

研究证明稀疏图的NAC着色存在性为NP完全问题，并提出高效算法以提高计算效率。

拉姆齐数的下界：$\mathrm{R}(6,8)\ge 135$ 和 $\mathrm{R}(8,10)\ge 345$

Lower bounds for Ramsey numbers: $\mathrm{R}(6,8)\ge 135$ and $\mathrm{R}(8,10)\ge 345$

本研究通过构造明确的红蓝着色方案，结合局部搜索等方法，改进了拉姆齐数R(6,8)和R(8,10)的下界，分别提升至135和345。

基于数字索引的q元SEC-DED码及其近汉明开销

Digit-Indexed q-ary SEC-DED Codes with Near-Hamming Overhead

本文提出了一种基于数字索引的q元SEC-DED码，通过近汉明开销和简洁结构实现高效解码，同时支持可变长度编码和三元距离4变种。

Homomorphism bounds of signed bipartite $K_4$-minor-free graphs and edge-colorings of $2k$-regular $K_4$-minor-free multigraphs

A signed graph $(G, Σ)$ is a graph $G$ and a subset $Σ$ of its edges which corresponds to an assignment of signs to the edges: edges in $Σ$ are negative while edges not in $Σ$ are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced otherwise. The unbalanced-girth of a signed graph is the length of a shortest unbalanced closed walk (if such a walk exists). A homomorphism of $(G,Σ)$ to $(H,Π)$ is a homomorphism of $G$ to $H$ which preserves the balance of closed walks. In this work, given a signed bipar…

欧几里得空间色数的新上界

New upper bounds for the chromatic numbers of Euclidean spaces

本文通过构造显式有理格和解析方法，将欧几里得空间色数上界改进为R^4≤43、R^5≤132等，反驳了相关猜想，并首次给出R^10低于3^n的界。