Extending the Bipartite Parking Space

We prove an analogue of a theorem of Berget and Rhoades about extending the parking space ${\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{k}}_n$ to an $S_{n+1}$-module ${\mathrm{S}\mathrm{l}\mathrm{i}\mathrm{m}}_n$. Specifically, we show that the $\mathit{\text{bipartite}}\mathit{\text{parking}}\mathit{\text{space}}$ ${\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{k}}_{K_{n,m}}$, which naturally comes with an $S_{n-1}\times S_m$ action, extends to an $S_n\times S_m$-representation ${\mathrm{S}\mathrm{l}\mathrm{i}\mathrm{m}}_{n,m}$. We then formulate a conjecture generalizing this statement to any simple graph.