Cycle-free partial orders and ends of graphs

Abstract

The relationship between posets that are cycle-free and graphs that have more than one end is considered. We show that any locally finite bipartite graph corresponding to a cycle-free partial order has more than one end, by showing a correspondence between the ends of the graph and those of the Hasse graph of its Dedekind–MacNeille completion. If, in addition, the cycle-free partial order is k-CS-transitive for some k ≥ 3 we show that the associated graph is end-transitive. Using this approach we go on to prove that, for infinite locally finite 3-CS-transitive posets with maximal chains of height 2, the properties of being crown-free and being cycle-free are equivalent. In contrast to this we show that the non-locally finite bipartite graphs arising from skeletal cycle-free partial orders each have only one end. We include a corrected proof of a result from an earlier paper on the axiomatizability of the class of cycle-free partial orders.