Bound the ratio for axis-parallel box intersection graphs

Determine whether the domination-packing ratio \(\gamma(G)/\rho(G)\) is bounded by a constant for intersection graphs of axis-parallel boxes in three dimensions.

Background

The paper explicitly singles out intersection graphs of axis-parallel boxes in three dimensions as a geometric graph class for which boundedness of the domination-packing ratio remains unresolved. It also notes that a bound of the form γ(G)≤(Δ(G)−1)ρ(G)+1\gamma(G)\le (\Delta(G)-1)\rho(G)+1 for this class would imply the subcubic conjecture.

References

The question of bounding the ratio $\gamma/\rho$ for graphs in many geometric graph classes remains open as well. For example, intersection graphs of axis-parallel rectangles in the plane or intersection graphs of axis-parallel boxes in three dimensions. Note that, using the result in , a bound of the form $\gamma(G)\le (\Delta(G)-1)\rho(G)+1$ in the latter class of graphs would imply Conjecture~\ref{conj:deg3}.

— On graph classes with constant domination-packing ratio  (2503.05562 - Bonamy et al., 7 Mar 2025) in Section 7, Conclusion