Dushnik-Miller dimension of contact systems of d -dimensional boxes

Mathew Francis Daniel Gonçalves 1
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : Planar graphs are the graphs with Dushnik-Miller dimension at most three (W. Schnyder, Planar graphs and poset dimension, Order 5, 323-343, 1989). Consider the intersection graph of interior disjoint axis-parallel rectangles in the plane. It is known that if at most three rectangles intersect on a point, then this intersection graph is planar, that is it has Dushnik-Miller dimension at most three. This paper aims at generalizing this from the plane to by considering tilings of with axis parallel boxes, where at most boxes intersect on a point. Such tilings induce simplicial complexes and we will show that those simplicial complexes have Dushnik-Miller dimension at most.
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Mathew Francis, Daniel Gonçalves. Dushnik-Miller dimension of contact systems of d -dimensional boxes. Electronic Notes in Discrete Mathematics, Elsevier, 2017, 61, pp.467-473. ⟨10.1016/j.endm.2017.06.075⟩. ⟨lirmm-01693192⟩



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