On the Word-Representability of Tensor Product Graphs

Abstract

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.