Abstract
Choi et al. (2016) introduced the notion of the partial order competition dimension of a graph and characterized the graphs with partial order competition dimension d in terms of homothetic regular (d−1)-simplices in Rd which are contained in the hyperplane {x∈Rd∣x⋅1=0}. In this paper, we introduce a useful notion “order types for two points in R3” and give an upper bound of the partial order competition dimension of a graph in terms of its chromatic number, which is achieved via constructing a matrix in a somewhat clever way, to study partial order competition dimensions of bipartite graphs and planar graphs.
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