The chromatic number of comparability 3–hypergraphs Natalia Garc´ ıa-Col´ ın Instituto de Matem´ aticas, UNAM natalia.garciacolin@im.unam.mx Amanda Montejano UMDI Facultad de Ciencias, UNAM amandamontejano@ciencias.unam.mx Deborah Oliveros Instituto de Matem´ aticas, UNAM dolivero@matem.unam.mx Abstract Beginning with the concepts of orientation for a 3–hypergraph and transitivity for an oriented 3–hypergraph, it is natural to study the class of comparability 3– hypergraphs (those that can be transitively oriented). In this work we show three different behaviors in respect to the relationship between the chromatic number and the clique number of a comparability 3–hypergraph, this is in contrast with the fact that a comparability simple graph is a perfect graph. Keywords: Perfection in 3–hypergraphs, transitivity in 3–hypergraphs, comparability 3–hypergraphs. 1. Introduction and motivation In [4] the authors introduce the concepts of orientation for a 3–hypergraph, tran- sitivity for an oriented 3–hypergraph, and define the class of comparability 3– hypergraphs as the class of non oriented 3–hypergraphs, which can be transitively oriented (precise definitions are provided in Section 2). These 3–hypergraphs are a natural generalization of (simple) comparability graphs (graphs which can be transitively oriented or, equivalently, graphs associated to a partially ordered set). Comparability graphs are well known to be perfect graphs. A graph is said to be perfect if all of its induced subgraphs have chromatic number equal to their clique Preprint submitted to Discrete Applied Mathematics March 5, 2014 arXiv:1402.5739v1 [math.CO] 24 Feb 2014