Contact graphs of boxes with unidirectional contacts
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This paper is devoted to the study of particular classes of geometrically defined intersection graphs. Those are contact graphs of axis parallel boxes in ℝ^d, where the intersection of any pair of boxes is parallel to a given hyperplane (among the d possible ones). C. Magnant and D.L. Martin showed that these graphs (already for d=3) have arbitrary large chromatic number, while being triangle free. We give several structural properties of these graphs, and those raise many questions. These graphs have the particular feature to serve as a counter-example for a conjecture of A. Atminas, V. Lozin, and V. Zamaraev.
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