On Lexicographic Product and Multi-Word-Representability

We investigate the relationship between the lexicographic product of graphs and their multi-word-representation number. We establish bounds on the multi-word-representation number $\mu$ for lexicographic powers and products. Specifically, if $G$ is a non-comparability graph, then $\mu(G^{[k]}) \le k$, whereas if $G$ is the union of two comparability graphs, then $\mu(G^{[k]}) = 2$. More generally, let $G_1$ and $G_2$ be graphs with $\mu(G_1) = k_1$ and $\mu(G_2) = k_2$. For their lexicographic product $H = G_1 \circ G_2$, we have $\mu(H) \le k_1 + k_2$. This bound is tight: $\mu(H) = k_1$ when $k_1 \ge k_2$ and $G_2$ is the union of $k_1$ comparability graphs. Furthermore, if $G_1$ and $G_2$ are minimal non-word-representable graphs, then $\mu(G_1 \circ G_2) \le 3$. Finally, we study the function $\tau(n)$, which measures the size of the largest word-representable induced subgraph guaranteed in every $n$-vertex graph. By constructing extremal graphs via lexicographic powers, we establish a sublinear upper bound, showing that $\tau(n) \le n^{0.86}$ for sufficiently large $n$.