关于张量积图的词可表示性

中文摘要 AI 辅助

词可表示图是一类可以用词表示的图，其中边和非边由这些词中字母的交替决定。张量积 $G \times H$（也称为直积或Kronecker积）是四种标准图积之一。Kitaev和Lozin的著作《Words and Graphs》（Springer, 2015）中的问题7.2.5提出了关于张量积的词可表示性的三个开放问题。尽管对词可表示图已有广泛研究，但张量积的词可表示性似乎未受到关注。本文不仅回答了Kitaev和Lozin在问题7.2.5中提出的2.5个问题，而且开启了一项系统性研究，聚焦于四个基本图族：轮图 $W_n$、完全图 $K_n$、圈图的Mycielskian图 $\mu_n$ 以及圈图的扩展Mycielskian图 $\mu'_n$。在我们的主要结果中，我们证明了对于任意图 $G$，$W_{2n} \times G$、$\mu_{2n} \times G$ 和 $\mu'_{2n} \times G$ 总是词可表示的；$K_n \times K_m$ 是词可表示的当且仅当 $\min\{n,m\} \leq 3$；并且包含 $W_{2n+1} \times W_{2m+1}$ 或 $\mu_{2n+1} \times \mu_{2m+1}$ 或 $\mu'_{2n+1} \times \mu'_{2m+1}$ 作为诱导子图的张量积 $G \times H$ 是不可词表示的。我们的证明利用了非词可表示性的遗传性质以及不可比较邻域的存在性。

英文摘要

Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. The tensor product $G \times H$ (also known as the direct product or Kronecker product) is one of the four standard graph products. Problem 7.2.5 in Kitaev and Lozin's book \emph{Words and Graphs} (Springer, 2015) raised three open questions regarding the word-representability of tensor products. Despite extensive research on word-representable graphs, the word-representability of tensor products seems to have received no attention. This paper not only answers 2.5 of the questions posed by Kitaev and Lozin in Problem 7.2.5, but also initiates a systematic study, focusing on four fundamental families: wheel graphs $W_n$, complete graphs $K_n$, the Mycielskian of the cycle graph $μ_n$, and the extended Mycielskian of the cycle graph $μ'_n$. Among our main results, we prove that $W_{2n} \times G$, $μ_{2n} \times G$, and $μ'_{2n} \times G$ are always word-representable for any graph $G$; that $K_n \times K_m$ is word-representable if and only if $\min\{n,m\} \leq 3$; and that tensor products $G \times H$ containing $W_{2n+1} \times W_{2m+1}$ or $μ_{2n+1} \times μ_{2m+1}$ or $μ'_{2n+1} \times μ'_{2m+1}$ as induced subgraphs are non-word-representable. Our proofs exploit the hereditary nature of non-word-representability and the presence of non-comparability neighbourhoods.

发表机构

• University of Hafr Al Batin（哈夫阿尔巴廷大学）
• University of Strathclyde（斯特拉斯克莱德大学）

机构由 AI 辅助整理，请以论文原文为准。