On graph products and multi-word-representability

Abstract

The multi-word-representation number $μ(G)$ of a graph $G$ is the minimum number of word-representable graphs whose union is $G$. We study the behavior of $μ$ under six standard graph products: the lexicographic, Cartesian, rooted, corona, tensor, and strong products. For the Cartesian and rooted products, we show that $μ(G_1 \square G_2)=μ(G_1 \diamond G_2)=\max\{μ(G_1),μ(G_2)\}$. For the corona product, we prove that $μ(G_1 \odot G_2)\le \max\{μ(G_1),μ(G_2)\}+1$, and we identify a condition under which equality holds. For the lexicographic product, we establish $μ(G_1 \circ G_2)\le μ(G_1)+μ(G_2)$, which reduces to $\max\{μ(G_1),μ(G_2)\}$ under a comparability cover condition on $G_2$, and we characterize the case when the lexicographic product of two minimal non-word-representable graphs has $μ=2$. For the tensor product $G_1 \times G_2$, we show $μ(G_1 \times G_2)\le \log_3(\min\{χ(G_1),χ(G_2)\})$. For the strong product $G_1 \boxtimes G_2$, we establish $\max\{μ(G_1),μ(G_2)\}\le μ(G_1 \boxtimes G_2)\le \max\{μ(G_1),μ(G_2)\}+\log_3(\min\{χ(G_1),χ(G_2)\})$. For lexicographic powers $G^{[k]}$, we prove that $μ(G^{[k]})\le k$ when $G$ is word-representable but not a comparability graph, and in general $μ(G^{[k]})$ is bounded by the comparability cover number of $G$. We further show that $G^{[k]}$ is word-representable if and only if $G$ is a comparability graph. As an application, we obtain a sublinear upper bound on the extremal function $τ(n)$, defined as the largest integer such that every $n$-vertex graph contains a word-representable induced subgraph of that size; in particular, $τ(8^k)\le 6^k$, implying $τ(n)\le n^{\log_8 6+ε}$ for large $n$.