Determine the optimal exploration-time bound for bounded-underlying-degree temporal graphs

Determine the largest possible exploration time of an always-connected temporal graph on n vertices whose underlying graph has maximum degree at most Δ, thereby closing the polynomial gap between the lower bound Ω(Δ n (1 + log(n/Δ))) and the upper bound O(min{n², n^{3/2}√(Δ log n)}), including the constant-degree case.

Background

The paper constructs always-connected temporal graphs with underlying maximum degree at most Δ whose exploration requires Ω(Δ n (1 + log(n/Δ))) time steps. The best cited upper bound is O(min{n², n{3/2}√(Δ log n)}), leaving a polynomial gap when Δ is polynomially smaller than n.

The gap remains even for constant underlying degree: the known lower bound is Ω(n log n), while the known upper bound is O(n{3/2}√(log n)). The authors also explain that their construction and assignment scheme are already tight for the graphs they build, so improving the general bound requires a different approach.

References

The two differ by a polynomial factor whenever $\Delta$ is polynomially smaller than $n$, and closing this gap is the main question, open even for constant degree: there the known bounds are $\Omega(n\log n)$ and $O(n{3/2}\sqrt{\log n})$, and our bound does not improve the lower one.

— Hiding a Vertex from the Temporal Explorer: A Lower Bound for Degree-Bounded Temporal Graphs  (2610.06228 - Carnevale, 5 Oct 2026) in Section 1, immediately before Section 1.1?; Section 'Open questions' (Section 9, \label{sec:conclusion})

We do not know whether any lower bound with a linear dependence on the degree survives two moves per step.

— Hiding a Vertex from the Temporal Explorer: A Lower Bound for Degree-Bounded Temporal Graphs  (2610.06228 - Carnevale, 5 Oct 2026) in Section 'Limits and variants of the construction', paragraph 'Several moves per time step'; reiterated in Section 'Open questions'

They conjecture that $O(\sqrt{\Delta_s}\,n)$ steps suffice with high probability when all trees have maximum degree $\Delta_s$.

— Hiding a Vertex from the Temporal Explorer: A Lower Bound for Degree-Bounded Temporal Graphs  (2610.06228 - Carnevale, 5 Oct 2026) in Section 'Related work', paragraph on random temporal graphs