Intersection graphs of maximal hypercubes

In this paper we consider cube graphs, that is intersection graphs of maximal hypercubes of graphs. In contrast to the related concepts of line graphs and clique graphs, we show that any graph is a cube graph of a (bipartite) graph. We answer a question of Bandelt and Chepoi (European J. Combin. 17 (1996) 113) by showing that dually chordal graphs are precisely cube graphs of graphs of acyclic cubical complexes. Similarly, we characterize classes of chordal graphs, Helly chordal graphs and doubly chordal graphs as cube graphs of certain subclasses of isometric subgraphs of hypercubes.