We show that there is a subdivision of $K_4$, the complete graph on 4 vertices, which is not the intersection graph of two dimensional axis-parallel boxes with some restrictions on the placement of these boxes. We dub this class restricted box graphs.

Our work arises in the study of $chi$-bounded classes of graphs, namely we consider a conjecture of Scott. It states that for any graph $H$, the class of graphs excluding all subdivisions of $H$ as induced subgraphs is $chi$-bounded. He showed this conjecture holds when $H$ is a tree, but a recent result of Pawlik et al. proves Scott's conjecture is false when $H$ is a (specific) subdivision of $K_5$. Restricted box graphs are not $chi$-bounded and we use this fact to show Scott's conjecture is false for a subdivision of $K_4$ by characterizing graphs whose 2-subdivision (the graph obtained by subdividing every edge twice) is a restricted box graph.

This is joint work with Jérémie Chalopin, Louis Esperet, Frantisek Kardos, Patrice Ossona de Mendez and Stéphan Thomassé.