Graph parking-space extension conjecture

For every finite simple graph $G$ with distinguished vertex $v$ , let $\operatorname{Aut}_v(G)\leq \operatorname{Aut}(G)$ be the subgroup fixing $v$ . Then the conjecture asserts that the restriction of the $\operatorname{Aut}(G)$ -representation $\mathrm{Slim}_G$ to $\operatorname{Aut}_v(G)$ is isomorphic to the parking-space representation $\mathrm{Park}_G$ : $\operatorname{Res}^{\operatorname{Aut}(G)}_{\operatorname{Aut}_v(G)}\mathrm{Slim}_G\cong \mathrm{Park}_G$ .

Progress summary

Partially solved

A new paper proves the conjecture for complete bipartite graphs, but the proposed extension to all simple graphs remains open.

The conjecture asks whether, for every simple graph $G$ with distinguished vertex $v$ , restricting $\mathrm{Slim}_G$ from $\operatorname{aut}(G)$ to $\operatorname{aut}_v(G)$ gives $\mathrm{Park}_G$ .

Known results

Berget and Rhoades proved the complete-graph case $G=K_{n+1}$ .
The conjecture is also known for trees and is described as easy for a single cycle; these cases are not attributed to named authors in the retrieved source.

August 2026 bipartite extension

Dora Woodruff’s paper, posted 15 August 2026, proves the complete bipartite case $G=K_{n,m}$ by constructing an explicit equivariant bijection. It formulates the general simple-graph statement as Conjecture 4.1 and reports computational tests, but leaves the arbitrary case open because suitable orbit representatives and a compatible partial order are not known.

Current status (as of August 2026): The conjecture is settled for complete graphs, complete bipartite graphs, trees, and a single cycle, while the case of arbitrary simple graphs remains open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Extending the Bipartite Parking Space — arXiv — Dora Woodruff

algebraic-combinatorics representation-theory

2 days ago