Characterizing graphs of critical pairs of layered generalized crowns

The generalized crown (Formula presented.) is a well-known family of bipartite graphs whose order dimension is given in terms of the parameters (Formula presented.) and (Formula presented.). In recent work, Garcia and Silva defined the notion of layering generalized crowns, producing multipartite posets called (Formula presented.) -layered generalized crowns, whose order dimension is easily determined using (Formula presented.), (Formula presented.), and (Formula presented.). This paper extends the authors’ prior work on characterizing the associated graphs of critical pairs of generalized crowns, by providing a new and concrete description of an infinite family of graphs arising from critical pairs of the (Formula presented.) -layered generalized crowns. Our main result gives a characterization of the adjacency matrices of these graphs. Through their associated posets with computable order dimension, these graphs have a strict upper bound on their chromatic number.