Quadratic local-complexity bounds for quasi-linear properties in general graphs

Determine whether the quasi-linear local-complexity lower bounds established for the domatic-number-at-most-two property, non-existence of a cubic subgraph, non-existence of a partition into k acyclic subgraphs, existence of a monochromatic triangle in every 2-edge-coloring, non-existence of a Hamiltonian cycle, and chromatic index equal to maximum degree plus one can each be improved to quadratic lower bounds in general graphs.

Background

The paper develops local hardness reductions that transfer lower bounds on certificate size between graph properties. Many of the resulting lower bounds are quasi-linear, of order Ω(n/log n), and are proved for bounded-degree graph classes. Because bounded-degree graphs admit a universal O(n log n)-bit certificate based on their adjacency lists, these bounds are close to optimal within those classes.

When the same properties are considered on general graphs, the paper notes that the quasi-linear lower bounds remain valid, but the universal upper bound rises to O(n²). The authors therefore explicitly identify as an open problem whether the lower bounds for the relevant properties can be strengthened from quasi-linear to quadratic in general graphs.