Recognition of interval k-graphs with an unspecified partition

Determine in polynomial time whether a given graph G admits a k-partition V_1,V_2,\dots,V_k such that G equipped with this partition is an interval k-graph.

Background

An interval k-graph is defined for a k-partite graph whose partite sets are given, with adjacency between vertices in different partite sets determined by intersection of their representing intervals. The paper establishes an O(|V(G)||E(G)|)-time recognition algorithm when the partition into k partite sets is part of the input.

The unresolved variant removes the supplied k-coloring or partition and asks whether some suitable partition exists. The paper notes that the authors have no conclusive indication that this problem is NP-complete and that prior work considered particular cases with k=2 and k=3. The open problem therefore concerns the computational complexity of recognizing graphs that admit some partition making them interval k-graphs.