De Bruijn Graph Homomorphisms and Recursive De Bruijn Sequences

On connection Bruijn-Erdös Theorem and symplectic geometry

In this article a family of recursive and self-similar matrices is constructed. It is shown that the Plücker matrix of the Isotropic Grassmannian variety is a direct sum of this class of matrices.

Arbitrary-length analogs to de Bruijn sequences

Let $\widetildeα$ be a length-$L$ cyclic sequence of characters from a size-$K$ alphabet $\mathcal{A}$ such that the number of occurrences of any length-$m$ string on $\mathcal{A}$ as a substring of $\widetildeα$ is $\lfloor L / K^m \rfloor$ or $\lceil L / K^m \rceil$. When $L = K^N$ for any positive integer $N$, $\widetildeα$ is a de Bruijn sequence of order $N$, and when $L \neq K^N$, $\widetildeα$ shares many properties with de Bruijn sequences. We describe an algorithm that outputs some $\widetildeα$ for any combination of $K \geq 2$ and $L \geq 1$ in $O(L)$ time using $O(L \log K)$ space.…

无零的德布鲁因环面：域论视角

De Bruijn Tori Without Zeros: A Field-Theoretic Perspective

本文提出一种基于有限域迹的代数构造方法，用于生成省略全零模式的非零德布鲁因环面，通过乘法独立生成元和线性映射实现高效计算。

A recursive approach for the enumeration of the homomorphisms from a poset $P$ to the chain $C_3$

Let ${\cal H}(P,C_3)$ be the set of order homomorphisms from a poset $P$ to the chain $C_3 = 1 < 2 < 3$. We develop a recursive approach for the calculation of the cardinality of ${\cal H}(P,C_3)$, and we apply it on several types of posets, including $P = C_3 \times C_3 \times C_k$ and $P = {\cal H}(C_k, C_3)$; for the latter poset $P$, we derive a direct formula for $\# {\cal H} ( P, C_3 )$.

可定向序列

Special orientable sequences

本文提出构造特殊可定向序列的方法，适用于Alhakim等人提出的两种递归方法，实现任意字母表大小和窗口大小的构造，序列周期接近最优。

Homomorphisms into loop-threshold graphs

Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and edges into small image graphs. For the image graph $H_{\text{ind}}$, the graph on two adjacent vertices, one of which is looped, each homomorphism from $G$ to $H_{\text{ind}}$ corresponds to an independent set in $G$. It follows from the Kruskal-Katona theorem that the number of homomorphisms to $H_{\text{ind}}$ is maximized by the lex graph, whose edges form a…